UNIVERSITY  FARM 


miffl 


1 


I 


THE  SHAWINIGAN  10,500  H.P.  TURBINE. 
Designed  and  built  by  the  I.  P.  Morris  Co.,  Phila.,  Pa. 
(Efficiency  at  official  test,  86%  at  full  load  ;  73^%  at  28^  load. 


.HYDEAULIC    MOTOB8 

WITH   RELATED    SUBJECTS 

INCLUDING 

CENTRIFUGAL  PUMPS,   PIPES, 
AND  OPEN  CHANNELS 

DESIGNED    AS 

A   TEXT-BOOK  FOR  ENGINEERING  SCHOOLS 


BY 

IRVING   P.  CHURCH,  M.C.E. 

Assoc.  AM.  Soc.  C.E. 

Professor  of  Applied  Mechanics  and  Hydraulics,  College  of  Civil  Engineering* 
Cornell  University 


FIRST    EDITION 
EIGHTH   THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:   CHAPMAN  &  HALL,  LIMITED 

1914 


UNIVERSITY  OF  CALIFORNIA 

LIBRARY 

BRANCH  OF  THE 
COLLEGE  OF  AGRICULTURE 


COPYRIGHT,  1905 

BY 

IRVING  P.  CHURCH. 


NOTE. — For  a  short  course,  to  include  Hydraulic  Motors  proper, 
the  following  paragraphs  may  be  selected,  viz.  :  §§  1-6,  14-59,  62-1043. 


PREFACE. 


BY  reason  of  the  great  increase  that  has  taken  place  of  recent  years 
throughout  the  world  in  the  utilization  of  water  power,  notably  in 
connection  with  the  electric  transmission  of  energy,  a  special  ana  grow- 
ing prominence  attaches  to  the  subject  of  Hydraulic  Motors  in  the 
curriculum  of  engineering  schools. 

In  the  preparation  of  the  following  pages,  as  forming  a  text-book 
on  this  important  branch  of  hydraulics  for  the  use  of  students  of  engi- 
neering, it  has  been  borne  in  mind  that  to  facilitate  the  acquirement 
of  clear  and  sound  ideas  on  the  mechanics  of  the  subject  is  the  first 
essential  of  such  a  book;  and,  as  greatly  assisting  to  this  end,  ample 
numerical  illustration  has  been  provided  in  direct  connection  with  the 
necessary  algebraic  treatment.  At  the  same  time,  it  is  believed  that 
sufficient  descriptive  matter  has  been  introduced,  relating  to  both  past 
and  present  construction  and  design,  to  make  the  treatment  a  fairly 
practical  one  for  its  purpose,  when  regard  is  had  to  the  limited  time 
available  for  this  subject  in  the  ordinary  course  of  study  at  an  engi- 
neering school. 

Some  attention  is  also  given  to  centrifugal  pumps  (so  much  improved 
of  recent  years)  and  other  allied  appliances;  and  to  special  problems, 
closely  connected  with  the  subject  of  water-power,  involving  pipes, 
weirs,  and  open  channels.  The  experiments  of  Joukovsky  on  water- 
hammer  are  presented  and  the  theory  of  this  phenomenon  is  developed. 

The  student  is  supposed  to  be  already  well  versed  in  the  part  of 
hydraulics  dealing  with  stationary  vessels  and  pipes,  as  set  forth  (for 
instance)  in  the  writer's  Mechanics  of  Engineering  (in  referring  to  which 
the  abbreviation  M.  of  E.  is  used). 

It  is  hoped  that  the  book  may  prove  useful  to  practising  engineers 
as  well  as  students;  in  which  connection  attention  is  called  to  the  dia- 
grams of  friction-heads  in  pipes  and  those  for  determining  Kutter's 
coefficients  for  open  channels.  These  have  been  especially  prepared  for 
the  present  work  and  will  be  found  in  the  Appendix. 


11  PREFACE. 

For  the  use  of  many  of  the  illustrations  appearing  in  the  pages  of 
this  work  the  writer  would  acknowledge  his  great  obligation  to  the 
following  technical  journals  and  engineering  firms: 

Engineering  News  (Figs.  9,  10,  37,  38,  39,  40,  89,  and  90), 

Cassier's  Magazine  (Figs.  52,  53,  54,  and  55). 

Kilburn,  Lincoln  and  Co.,  Fall  River,  Mass. 

I.  P.  Morris  Co.,  Phila. 

Risdon-Alcott  Turbine  Co.,  Mount  Holly,  N.  J. 

Pelton  Water-wheel  Co.,  San  Francisco. 

Abner  Doble  Co.,  San  Francisco. 

James  Leffel  Co.,  Springfield,  Ohio. 

Allis-Chalmers  Co.,  Milwaukee,  Wis. 

Platt  Iron  Works  Co.,  Dayton,  Ohio  (Victor  Turbine). 

Dayton  Globe  Iron  Works,  Dayton,  Ohio  (New  American  Turbine). 

Lombard  Governor  Co.,  Ashland,  Mass. 

De  Laval  Steam  Turbine  Co.,  Trenton,  N.  J.  (Centrifugal  Pumps). 

Lawrence  Machine  Co.,  Lawrence,  Mass.  (Centrifugal  Pumps). 

Irvin  Van  Wie,  Syracuse,  N.  Y.  (Centrifugal  Pumps). 

Henry  R.  Worthington,  New  York  (Water-motor  Pump). 

Columbia  Engineering  Works,  Portland,  Oregon  (Hydraulic  Ram). 

Goulds  Manufac.  Co.,  Senaca  Falls,  N.  Y.  (Hydraulic  Ram). 

CORNELL  UNIVERSITY, 
ITHACA,  N.  Y.,  Sept.  1905. 


CONTENTS. 


CHAP.  I.    GENERAL    CONSIDERATIONS,    AND    PRINCIPAL    TYPES 

OF  MOTORS. 

§§  1-13.  Water-power.  Gravity,  Pressure,  and  Inertia,  Types 
of  Water-motor.  General  Theorem  for  Power  of  any 
Water-motor;  also  for  Pump 1-21 

CHAP.  II.    GRAVITY  MOTORS. 

§§  14-30a.  Overshot  Wheels.  Power  due  to  Weight  and  to 
Impact.  Breast  wheels.  Back- pitch  and  Sagebien  Wheels. 
Undershot  Wheels.  Current  Wheels  and  Poncelet  Under- 
shots.  Gearing 22-38 

CHAP.  III.     PRELIMINARY  THEOREMS,  FUNDAMENTAL  TO  THE  THEORY 
OP  TURBINES  AND  CENTRIFUGAL  PUMPS. 

§§31-42a.  Theorems  A,  B,  and  C.  "Angular  Momentum." 
Power  of  Turbine  in  Steady  Operation.  Friction  con- 
sidered. Bernoulli's  Theorem  for  a  Rotating  Casing,  etc. 
"Turbine  Pump" 39-61 

CHAP.  IV.    IMPULSE  WHEELS. 

§§43-57.  Pressure  of  Free  Jet  on  Solid  of  Revolution,  Fixed 
or  Moving.  Pelton,  Doble,  and  Cascade  Impulse  Wheels; 
and  their  Regulation.  Girard  Impulse  Wheels 62-82 

CHAP.  Y.  TURBINES  AND  REACTION-WHEELS. 
•§§  5S-94a.  The  Reaction-wheel.  Development  of  the  Turbine. 
Description  and  Theory  of  the  Fourneyron  Turbine. 
Theory,  with  Friction.  Classification  of  Turbines.  Radial- 
flow,  Axial-flow,  and  Mixed-flow  Turbines.  Francis  and 
Jonval  Turbines.  Turbines  at  Niagara  Falls.  The  Draft- 
tube  and  the  Diffuser.  American  Turbines.  History. 
General  Theory  of  the  Reaction  Turbine.  Guide-blades 
and  Turbine- vanes.  Scheme  of  Computation 82-148 

CHAP.   VI.    TESTING  AND  REGULATION  OF  TURBINES. 
§§  95-104a.  Friction-brake   and   Hook   Gauge.     The   Holyoke 
Testing-flume.     Test  of  Tremont  Turbine;  with  Discussion. 
Regulating-gates  for  Turbines.    Turbine  Governors;  King, 
Snow,  Lombard,  etc 149-167 

CHAP.  VII.     CENTRIFUGAL  AND  "  TURBINE  "  PUMPS. 
,§§  105-114.  Description  and  Theory.     "Impending  Delivery" 
Pumps  with,  and  without,  Diffusion  Vanes.     Multi-stage 
Turbine  Pumps 168-187 


W  CONTENTS. 

CHAP.  VIII.    PIPES,  WEIRS,  AND  OPEN  CHANNELS. 

PAOB» 

§§  115-146.  Friction-head  in  Pipes.  Diagrams.  Hydraulic 
Grade-line.  Branching  Pipes.  Supply-pipes  for  Motors. 
Water-hammer  in  Pipes.  Joukovsky's  Experiments.  Open 
Channels.  Kutter's  Formula  and  Coefficient.  Diagrams 
for  Coefficient.  Backwater  due  to  Weirs;  Height  and 
Amplitude.  Rafter's  Experiments.  Submerged  Weirs. 
Standing  Waves 188-239 

CHAP.  IX.     PRESSURE-ENGINES,  ACCUMULATORS,  AND  HYDRAULIC 

RAMS. 

§§  147-163.  Pressure -engines;  Worthington,  Brotherhood, 
Schmidt,  etc.,  Accumulators.  Leather  and  Hemp  Pack- 
ing. Hydraulic  Rams.  Experiments  on  Rams.  Effi- 
ciency and  Rules  for  Parts.  Special  Designs;  Rife,  Pear- 
sail,  Mead,  Phillips,  etc.  Hydraulic  Air-compression 240-269" 

APPENDIX  OF  DIAGRAMS  AND  TABLES. 
INDEX. 

BIBLIOGRAPHY  OF   HYDRAULIC   MOTORS. 


Daugherty:  Hydraulic  Turbines.     New  York,  1913. 

Thurso:  Modern  Turbine  Practice  and  Water-power  Plants.  New- 
York,  1905. 

Wagenbach:  Neuere  Turbinen-Anlagen.     Berlin,  1905. 

Mueller:  Die  Francis-Turbinen.     Hannover,  1901. 

Bodmer:  Hydraulic  Motors,  etc.     London,  1889  and  1895. 

Rateau:  Traite  des  Turbo-machines.     Paris,  1900. 

Weisbach:  Hydraulic  Motors  (Prof.  Du  Bois,  transl.).     New  York,  1877. 

Inness:  The  Centrifugal  Pump,  Turbines,  and  Water-motors.  London,, 
1893. 

Meissner:   Die  Hydraulik  und  Hydraulischen  Motoren.     Jena,  1878. 

Buchetti:   Les  Moteurs  Hydrauliques  Actuels.     Paris,  1892. 

Blaine :  Hydraulic  Machinery.     London,  1897. 

Robinson:  HydrauJvi  Power  and  Hydraulic  Machinery.  London  (2d 
ed.),  1893. 

Marks:  Hydraulic  Power  Engineering.     London,  1900. 

Zeuner:  Theorie  der  Turbinen.     Leipsic,  1899. 

Herrmann:  Graphische  Theorie  der  Turbinen  und  Kreiselpumpen.. 
Berlin,  1887. 

Bjorling:  Water  or  Hydraulic  Motors.     London,  1894. 

Frizell:  Water-power.     New  York,  1901. 

Francis:  Lowell  Hydraulic  Experiments.     Boston,  1855. 

Chapters  on  hvdraulic  motors  may  also  be  found  in: 
Cotterill's  Applied  Mechanics; 
Merriman's  Hydraulics; 
Bovey's  Hydraulics; 
Rankine's  Steam-engine ; 
Unwin's  Hydromechanics  (Encyc.  Britann.). 


NOTATION  AND  CONSTANTS. 


The  Greek  letters  a,  /?,  £,  0,  £,  A,  /*,  and  0  are  used  for  angles  [d  also 
for  a  ratio  (pp.  25  and  32) ;  £  for  a  coefficient  (pp.  105,  174,  and 
192);  and  j*  as  a  coefficient  in  weir  formulae];  ij  for  efficiency;  -  and 
-  for  the  ratio  3.1416. 

7  (gamma)  is  the  weight  of  a  unit  of  volume  of  fresh  water,  viz.  62.3 
Ibs.  per  cub.  ft.  at  62°  Fahr.  (or  0.03604  Ibs.  per  cub.  inch) ;  but 
62.5  (or  1000-r- 16)  is  quite  accurate  enough  for  ordinary  hydraulic 
problems.  (Sea -water  weighs  64  Ibs.  per  cub.  ft.) 

*t»  (omega)  is  angular  velocity  of  a  rotating  body  (e.g.,  radians  per 
second;  in  which  case  revs,  per  sec.  would  be  to+2x}. 

A  is  Kutter's  coefficient  (p.  215). 

b  is  the  height  of  the  (ideal)  water-barometer.  For  a  pressure  of  one 
standard  atmosphere  (14.70  Ibs.  per  sq.  inch  or  2117  Ibs.  per  sq.  ft.) 
b  is  34  lineal  ft.;  corresponding  to  a  mercury  column  of  30  inches, 
nearly,  (29.95  in.).  In  any  actual  case  the  value  of  b  depends  on 
the  weather  and  the  elevation  above  sea-level.  For  example,  at 
6000  ft.  above  sea-level  it  might  be  about  27  ft. 

c,  ci,  cn,  etc.,  are  relative  velocities  of  water,  on  pp.  37,  38,  57-61,  and 
73-187. 

c  is  an  absolute  velocity  (of  water)  on  pp.  1-35  and  62-70. 

v,  v',  Vi,  etc.,  are  linear  velocities  of  points  of  a  revolving  body  (turbine) 
on  pp.  1-187.  v  and  c  =  mean  velocity  of  water  in  pipe  or  channel 
on  pp.  188-237. 

w,  wi,  wn,  etc.,  are  absolute  velocities  of  water  passing  through  a  motor; 
but  =  wetted  perimeter,  p.  229. 

h,  H,  y,  and  z  are  used  for  vertical  heights. 

/  is  the  coefficient  of  fluid  friction,  pp.  188-237. 

Z  is  a  length. 

F  is  the  area  of  a  cross  section  of  the  passageway  of  a  turbine,  or  of  pipe 
or  channel. 

u  is  "velocity  of  whirl"  (p.  48). 

V  is  "velocity  of  flow"  (p.  172). 

v 


vi  NOTATION  AND    CONSTANTS. 

L  is  power  (ft.-lbs.  per  sec.,  e.g.),  or  rate  of  work. 

Q  (rarely  q)  is  the  volume  of  water  flowing  per  unit  time  in  steady  flow 
(e.g.,  cubic  ft.  per  sec.;  gallons  per  minute). 

H.P.  =  horse-power  (=  ft.-lbs.  per  sec.  power-f-550). 

p  is  unit  pressure;  e.g.,  Ibs.  per  sq.  in.  (but  an  acceleration  on  p.  40, 
and  the  height  of  a  weir  on  p.  223). 

P  is  a  force  (or  total  pressure)  (Ibs.) ;  and  R,  or  R',  a  resistance  (i.e.,  a 
force)  (Ibs.).  R  is  "hydraulic  radius"  on  pp.  215,'  216. 

r,  n,  etc.,  are  used  for  radii;  d  for  diameter  (also  depth). 

s  is  the  slope  of  the  water  surface  in  an  open  channel,  p.  214. 

G  is  the  total  weight  of  a  body. 

g  is  the  acceleration  of  gravity.  In  the  temperate  zones  we  may  use, 
for  all  ordinary  problems  in  hydraulics,  the  value  32.2  (for  the 
English  foot  and  second  as  units)  (or  386.4  for  the  inch  and  second), 
as  sufficiently  accurate;  the  error  involved  being  only  a  small 
fraction  of  one  per  cent.  Near  the  equator  0=32.09  at  sea-level, 
and  32.06  at  10,000  ft.  elevation.  It  is  32.18  at  London  and  32.15 
at  Baltimore;  32.26  at  the  pole,  sea-level. 

One  U.  S.  gallon  of  fresh  water  (see  Conversion  Scales,  in  Appendix) 
weighs  8.34  Ibs.  at  ordinary  temperatures  and  has  a  volume  of 
231  cub.  in.  (or  0.1336  cub.  ft.).  One  cub.  ft.  contains  7.48  U.  S. 
gallons.  (N.B.  This  gallon  measure  is  in  common  use  in  this 
country  and  must  not  be  confused  with  the  English,  or  Imperial, 
gallon,  which  contains  277.27  cub.  in.  An  English  gallon  of  fresh 
water  weighs  10  Ibs.) 

GREEK  ALPHABET. 

Letters.  Names.  Letters.  Names. 

A  a  Alpha  N  v  Nu 

B  ft  Beta  g  £  Xi 

r  y  Gamma  O  o  Omicron 

A  S  Delta  Hit  Pi 

E  e  Epsilon  P  p  Rho 

Zeta  2<rs  Sigma 

H  -n  Eta  T  T  Tau 

©  fl  6  Theta  T  v  Upsilon 

I    z  Iota  $  0  phi 

K  K  Kappa  X  x  Chi 

A  A  Lambda  W  fy  Psi 

M  J*  Mu  a  GO  Omega 


HYDRAULIC  MOTORS. 


CHAPTER  I. 
GENERAL  CONSIDERATIONS  AND  PRINCIPAL  TYPES  OF  MOTORS. 

i.  Water-power. — The  descent  of  water  from  a  higher  to  a 
lower  level,  through  a  properly  designed  machine,  suitably 
regulated  as  to  speed  by  the  imposing  of  certain  resisting  forces 
to  prevent  acceleration  of  the  motion  of  the  machine,  may  be 
made  the  means  of  furnishing  certain  pressures  or  ''working 
forces/'  acting  at  different  parts  of  the  machine,  by  whose 
action  a  steady  or  uniform  motion  of  the  machine  may  be  kept 
up  notwithstanding  the  presence  of  the  resisting  forces.  In 
such  a  case  the  continuous  overcoming  of  the  resistances  is  said 
to  be  accomplished  by  Water-power,  and  the  machine  is  called 
a  Hydraulic  Motor. 

If  the  resultant  pressure  of  the  water  on  the  machine  or 
"  motor  "  is  P  Ibs.,  and  its  point  of  application  travels  uni- 
formly at  the  rate  of  v  ft.  per  second  in  the  direction  of  the  force 
P,  then  the  power  of  the  water  exerted  on  the  machine  is  the 
product  Pv  ft. -Ibs.  per  second,  (which  divided  by  550  gives 
Horse-power;)  and  if  there  is  but  one  resistance,  of  R'  Ibs., 
applied  to  the  motor,  and  its  point  of  application  is  forced 
to  travel  backwards  (backwards  as  regards  the  direction  of 
pointing  of  the  resistance  #0  at  the  rate  of  i/  ft.  per  second, 
the  power  thus  expended  is  R'tf  ft.-lbs.  per  second;  and  we 
have  the  equality 

(1) 


2  HYDRAULIC    MOTORS.  §  2. 

since  the  R'  is  supposed  to  have  such  a  value  that  the  motion 
of  the  machine  is  not  accelerated.     (See  §  146,  M.  of  E.) 

2.  Motors   of  the   Gravity,   Pressure,   and  Inertia  Types. — 
The  continuous  maintenance  of  this  working  force,  or  pressure, 
P,  of  the  water  against  the  motor  is  due  generally,  in  the  last 
analysis,  to  gravity,  i.e.,  to  the  weight  of  the  water;  but  it  is 
not   necessarily  due    to   the  weight  of  the    portions  of    water 
in  actual  contact  with  the  motor;  such  is  the  fact,  indeed,  (or 
nearly  so,)  in  the  case  of  motors  carrying  detached  bodies  of 
water  in  buckets,  and  these  may  be  called  pure  gravity  motors; 
but  in  the  case  of  pressure  engines,  with  slowly  moving  pistons, 
the  pressure  is  kept  up  by  communication  with  a  distant  and  large 
body  of  water;   while  with  turbines,  and  with  motors  utilizing 
a  "free  jet  "  (i.e.,  a  jet  in  the  open  air)  of  high  velocity,  the 
pressure  is  occasioned  by  causing  the  liquid  to  flow  through 
channels  or  against  surfaces  of  the  motor  in  such  a  way  that 
its  absolute  velocity  is  diminished  by  the  constraint  which  the 
parts  of  the  motor,  if  properly  designed,  exert  upon  its  motion. 
This  change  of  absolute  velocity  is  usually  accompanied  by  a 
gradual  change  of  direction,  to  avoid  waste  of  energy.     These 
latter  may  be  called  Inertia  Motors. 

(In  the  case  of  an  Inertia  motor  the  water  usually  gains  its 
initial  absolute  velocity,  at  entrance  of  the  motor,  through  the 
previous  action  of  gravity,  though  in  some  cases  this  velocity 
may  be  due  to  the  action  of  a  pump  driven  by  steam  or  other 
power.)  We  may  therefore  distinguish  between  Gravity  Motors, 
Pressure  Motors,  and  Inerlia  Motors  for  Kinetic  Motors); 
though  some  belong  to  more  than  one  of  these  categories,  as 
will  be  seen. 

3.  Efficiency. — If  a  motor  could  be  so  designed  (and  regu- 
lated) as  to  use  the  full  supply,  Q  cu.  ft.  per  second,  of  a  stream, 
and  also  the  full  "head,"  h  (feet),  or  difference  of  level  between 
the  surface  of  the  water  in  the  "  head- water  "  (or  pond)  and 
4  c  tail- water  "  (or  pool  where  the  water  flows  away,  below  the 
motor),  the  maximum  theoretical  water-power  would  be  equiva- 
lent to  a  working  force  equal  to  a  weight  of  Qf  Ibs.  (7-  being  the 
weight  of  one  cubic  foot  of  water)  working  through  h  ft.,  in 


§4, 


GENERAL    CONSIDERATIONS. 


each  second  of  time;  i.e.,  equivalent  to  Qj-h  ft.-lbs.  per  second; 
but  the  useful  power,  R'v',  accomplished  by  a  motor  at  its 
very  best  is  always  less  than  this,  on  account  of  various  kinds 
of  friction  and  because  the  water  itself  usually  leaves  the  motor 
with  a  certain  amount  of  velocity,  thus  carrying  away,  un- 
utilized, a  corresponding  amount  of  kinetic  energy,  each  second. 
The  ratio  of  the  power  usefully  expended,  viz.,  R'v',  to 
the  full  theoretical  maximum,  Qfh,  is  called  the  Efficiency  of 
the  motor  and  will  be  denoted  by  the  symbol  y  (pronounced 
ay-tah) ;  that  is, 

(2) 


4.  Example  of  a  Gravity  Motor. — A  succession  of  buckets 
on  an  endless  chain,  confined  in  their 
motion  to  a  vertical  plane  (the  chain 
passing  over  two  sprocket-wheels  H 
whose  axles  revolve  in  firm  hori- 
zontal bearings)  constitutes  a  nearly 
pure  gravity  motor.  See  Fig.  1. 
Each  bucket  as  it  moves  down 
receives  water  at  the  point  A  and 
loses  its  contents  at  B.  A  resistance 
Rf  (tension  in  a  rope,  e.g.,  winding 
up  on  drum  at  C,  being  of  sufficient 
value,  we  have  a  uniform  velocity  v 
of  bucket,  that  of  the  rope  being  v'° 
It  will  be  seen  from  the  figure  that 
the  height  h\,  from  A  to  B,  is  a  little 
less  than  that,  h,  from  head-water 
surface  H  to  tail-water  surface  at  T. 
Since  the  motion  of  a  bucket  while 
holding  water  is  uniform  and  recti- 
linear, the  resultant  pressure  of  the 
water  within  it  upon  the  bucket  is  equal  to  the  weight  of 
its  contents,  which  we  may  call  G  Ibs. 

If  wre  consider  the   buckets,   sprocket-wheels,   chain,   and 
drum  as  a  collection  of  rigid  bodies  forming  a  machine,  and 


Fio.  1. 


4  HYDRAULIC    MOTORS.  §  4, 

apply  the  method  of  "  Work  and  Energy  "  (see  pp.  149- 
153,  M.  of  E.),  we  note  that  there  is  a  working  force  G  acting; 
on  each  of  the  n  full  buckets  on  the  left;  that  R'  is  the  only 
resistance  (axle  frictions  are  here  neglected)  ;  *  and  that  the- 
reactions  at  the  bearings  are  neutral  forces  in  this  connection;, 
and  also  that  there  is  no  change  in  the  kinetic  energy  of  the 
moving  masses  of  the  collection  (by  hypothesis)  from  second 
to  second.  Hence,  considering  the  space  of  one  second  of  time, 
we  have 

R'vf  .........     (3) 


Let  now  £  =  time  for  a  bucket  to  pass  from  A  to  B\   then. 
v  =  hi  +  t  and 


But  nG  Ibs.  of  water  -v-£=lbs.  passing  per  second,  =  volume 

nG 
per  second  X?-,  i.e.,  —  =  Qr, 

0 

so  that  we  have  finally 

.    (4) 


Evidently,  with  greater  perfection  of  design  and  operation  the- 
quantity  Q?hi  could  approach  Qfh  but  could  not  exceed  it; 
hence  Qj-h  is  called  the  full  theoretical  power  of  the  "  mill- 
site,"  and  we  have  for  the  efficiency  the  ratio  (as  before  defined) 


Numerical  Example.—  With  Q  =  2  cub.  ft.  per  sec.  and  h  =  20  ft., 
we  have,  using  the  ft.-lb.-sec.  system  of  units,  0^  =  2X62.5X20- 
=  2500  ft.-lbs.  per  second,  maximum  theoretical  power.  Hence 
if  the  bucket-motor  is  so  'designed  as  to  have  an  efficiency  of 
80  per  cent  and  the  velocity  of  cable  at  C  is  desired  to  be  v'  =  2 
ft.  per  second,  we  may  put  R'v'  =  0.80  X  2500  and  obtain  R'  =  1000 
Ibs.  tension,  as  the  resistance  that  could  be  overcome  by  the 
motor  at  that  speed,  in  steady  motion.  If  the  velocity  of  the 
buckets  themselves  is  kept  at  the  value  (say)  v  =  3  ft.  per 

*  The  weight  of  the  wheels  and  buckets  is  a  neutral  force,  since  their 
cent°r  of  gravity  neither  sinks  nor  rises. 


§5. 


GRAVITY    TYPE    OF    MOTOR. 


5 


second,  the  radius  of  the  drum  C  must  be  made  two  thirds  of 
that  of  the  upper  sprocket-wheel. 

Strictly,  the  pressure  on  a  bucket  during  filling  at  position 
A  is  a  little  greater  than  the  weight  of  the  water  in  it  at  any 
stage  of  the  filling;  again,  both  the  filling  and  the  emptying 
of  any  bucket  are  gradual.  These  facts  are  neglected  at  present, 
for  simplicity,  but  will  be  considered  later. 

5.  Buckets  Moving  in  a  Circular  Path. — If  the  buckets  are 
firmly  attached  to  the  rim  of  a  single  rigid  wheel  (revolving  in 
a  vertical  plane)  and  thus  constitute  a  vertical  water-wheel, 
the  resultant  pressure  on  a  bucket  of  the  water  in  it  is  not  equal 
to  the  weight  of  that  water  during  uniform  motion,  but  the 
•effect  as  to  power  is  the  same;  that  is,  we  shall  have  Qfhi  =  Rfvf 
as  before;  hi  being  the.  vertical  distance  from  the  point  of 
filling  to  that  of  emptying. 


FIG.  2. 


To  prove  this,  in  simple  fashion,  consider  (Fig.  2)  a  heavy 
ball  of  weight  G  Ibs.,  resting  against  a  plate  ai  parallel  to  the 
radial  arm  nC,  and  upon  another  plate,  ic,  perpendicular  to 
the  same;  both  plates  perpendicular  to  the  vertical  plane  of 
the  paper.  The  arm  oC  and  plates  are  rigidly  fastened  to 
drum  CD  and  axle  K.  There  is  a  resistance  R'  acting  at  edge 
of  drum  (tension  in  a  rope,  say).  The  rigid  body  ainCK  is 
rotating  uniformly,  the  ball  with  it,  counter-clockwise,  on  axle 
K,  in  (vertical)  plane  of  paper. 

Let  #  =  the  angle  between  the  arm  nC  and  the  horizontal 
at  this  instant  (or  between  the  plate  ic  and  the  vertical).  Let 
the  reaction  of  plate  ai  against  the  ball  be  a  force  T'  Ibs. ;  that 


6  HYDRAULIC    MOTORS.  §  6. 

of  plate  in,  Nf  Ibs.  The  only  other  force  acting  on  the  ball 
is  that  of  the  earth,  i.e.,  its  weight,  G.  The  motion  of  the 
center  of  the  ball  being  curvilinear,  in  the  arc  of  a  circle  whose 
radius  is  r,  and  having  a  uniform  velocity  v,  in  that  curve^ 
we  have  [from  p.  76,  M.  of  E.] 

2Xtang.  compons.)  =0,     or    (7cos  6  —  77>  =  0; 
and  ^(norm.  compons.)  =  (G  +  g)v2  +  r; 

i.e.,  GawO-N'  =  (G  +  g)(vi  +  r). 

Hence  the  value  of  the  pressure  T'  against  the  ball  is  G  cos  6T 

Gv2 

while  that  of  N'  is  not  G  sin  0,  but  is  G  sin  0 . 

gr 

However,  when  we  apply  the  principle  of  work  and  energy 
to  the  rigid  body  anD  for  the  very  -short  time  interval,  dt,  in 
which  point  o  passes  to  of,  describing  a  path  of  length  ds,  while 
a  short  length  ds' ,  of  rope,  winds  up  on  the  drum,  dealing  now 
with  the  equals  and  opposites  of  N'  and  T',  we  have,  the  motion 
being  uniform,  T' .  ds  +  N'  X  zero  =  R'ds'. 

But  Tf  =  G  cos  6  and  ds  cos  0  =  OH  =  dh,  =  vertical  descent 
of  the  center  of  gravity  of  the  ball  in  time  dt,  and  hence 

Gdh  =  R'ds', (1) 

the  same  as  would  have  been  found  in  the  foregoing  case  of 
the  bucket-motor  for  the  time  dt  (with  nG  in  place  of  the 
present  G);  and  therefore,  for  a  complete  second,  we  should 
have 

Qfhi  =  R'v';    (see  later,  in  the  overshot  wheel.)      .     (2) 

6.  Simple  Pressure  Engine. — Fig.  3.  Here  we  consider  a 
single  stroke,  from  left  to  right,  of  a  piston  of  area  F.  sq.  ft., 
under  water  pressure  on  both  sides,  from  the  tanks  H  (head- 
water) and  T  (tail- water) ,  whose  surfaces  are  h  ft.  apart, 
vertically.  The  motion  is  slow  and  uniform,  acceleration 
being  prevented  by  the  action  of  a  suitable  resistance  R'  Ibs, 
against  the  piston-rod  (whose  sectional  area  is  small  com- 
pared with  that,  F,  of  the  piston).  The  unit-pressure  on  the 
left  face  of  the  piston  is  pm  (Ibs.  per  sq.  in.),  a  little  kss  than  the 
hydrostatic  pressure  due  to  the  depth  HE  (plus  the  outside 


§6. 


PRESSURE   TYPE    OF    MOTOR. 


atmospheric  pressure  pa)  on  account  of  the  loss  of  head  at 
entrance  E  of  communicating  orifice,  or  port.  Similarly,  the 
unit-pressure  at  mf ',  on  the  right-hand  face,  is  pm>,  a  little  greater 
than  that  due  to  the  vertical  depth  Tmf  (plus  atmosphere)^ 
If  piezometric  tubes  A  and  B,  open  to  the  air,  are  provided 
in  the  sides  of  the  cylinder,  as  shown,  the  heights,  y  and  y', 
of  the  stationary  water  columns  in  them  above  the  level  mm' 
will,  with  atmospheric  pressure  added,  measure  the  pressures 


FIG.  3. 


pm  and  pm*.  The  motion  of  the  water  is  assumed  to  be  a  "  steady 
flow/7  so  that  these  water  columns  do  not  fluctuate  in  height. 
Hence  we  write 

and     m'  =  a+'' 


so  that  for  steady  motion  the  value  of  the  resistance  Rf  should 
be 


Hence,  the  work  done  upon  the  resistance  in  one  stroke 
being  R's,  we  have  Fj-shi  =  R's  ........  (3) 

But,  if  n  strokes  are  made  in  a  unit  of  time,  say  one  second, 
(provision  being  made,  by  means  of  valves  and  of  air-vessels 
and  by  the  employment  of  more  than  one  cylinder  and  piston, 
for  the  maintenance  of  continuous  operation  and  of  a  practically 
"  steady  flow/')  we  have 

Work  per  second,  i.e.,  the  power  =  wFrshi,  =  R'(ns).    .     (4) 
Now  nFs  =  ihe  volume  of  water  used  per  unit  of  time,  =Q, 


8  HYDRAULIC    MOTORS.  §  6. 

and  ns  =  the  velocity,  v',  of  the  point  of  application  of  the 
resistance  R'  in  the  direction  of  the  latter  (in  general,  pro- 
jected on  the  line  of  action  of  the  resistance) ;  whence  the  power, 
L,  of  the  motor  may  be  written 

L,  =  Qrhl}  =  R'v' (5) 

It  is  evident  that  hi  can  never  quite  equal  h,  though  it  may 
be  made  to  approach  it  quite  closely  in  the  case  of  this  kind  of 
motor;  that  is,  as  before,  the  ideal  maximum  power  is  Qfh, 

R'v' 
and  the  efficiency  =  —7. 

We  here  note  that  in  passing  from  position  E  to  the  point 
where  it  leaves  the  motor  the  water  has  not  been  subjected  to 
any  notable  change  in  velocity,  nor  in  vertical  position;  that 
is,  that  between  E  and  m'  there  has  been  no  change  in  kinetic, 
nor  in  potential,  energy,  but  that  there  has  occurred  a  great 
change  in  the  internal  fluid  pressure;  so  that  this  kind  of 
motor  is  sometimes  described  as  acting  by  the  surrender  on 
the  part  of  the  water  of  some  of  the  " pressure  energy"  pos- 
sessed when  in  position  m.  But  it  should  be  remembered  that 
these  phrases  are  arbitrary  and  artificial,  being  employed 
simply  for  convenience.  Some  authors  use  the  word  potential 
energy  as  including  pressure  energy.  Others  would  say  that 
the  potential  energy  contained  in  the  water  at  H  has  been 
converted  into  the  form  of  pressure  energy  at  m,  since  no  con- 
version into  energy  of  motion  (i.e.,  into  kinetic  energy)  has 
taken  place  at  that  stage. 

Numerical  Example. — If  a  water-pressure  engine  is  working 
•steadily  with  a  piston  speed  of  v'  =  8  in.  per  second,  the  diameter 
of  piston  being  11.72  in.;  with  value  of  7^  =  70  ft.,  and  of  Ai=64 
ft. ;  we  have  for  the  power  obtained  (denote  it  by  L) 

^  |2-|x62.5x64, 
=  2000  ft.-lbs.  per  second,     or     3.63  horse-power. 

The  quantity  of  water  used  per  second  is  Q=--Fv'  =  Q.5  cu.  ft. 
per  second,  and  the  thrust  in  the  piston-rod  Rf  is  L  +  i/  or 


§7. 


INERTIA  TYPE    OF   MOTOR. 


9 


2000  -f-  0.666  =  3000  Ibs.     (If  we  neglect  the  friction  on  edges 
of  piston  and  in  stuffing-box).    The  efficiency  TJ  is 

flV  2000 

*     Qrh     0.5X62.5X70 

or  nearly  92  per  cent.     This  might  be  obtained  more  eas^y  by 
putting 

7?  =  /ii-rft;    or    64-v-TO;     =0.914. 

7.  A  Simple  Inertia,  or  Kinetic,  Motor. — It  has  already  been 
proved  in  §  566  of  the  M.  of  E.  (and  will  also  be  shown  later  in 
this  work)  that  if,  by  provision  of  a  proper  resistance  R', 
the  speed  of  the  cups  of  an  impulse  water-wheel,  such  as  a 
Pelton  or  Doble  wheel,  be  regulated  to  a  value  of  about  one 
half  that  of  the  water  in  the  free  "jet  "  (or  jet  in  the  open  air) 
issuing  from  a  nozzle  and  opera- 
ting upon  the  cups  in  succession, 
a,  maximum  power  is  obtained; 
that  is,  we  have  a  maximum 
value  for  the  product,  Pv,  of  the 
(mean)  tangential  force  (work- 
ing force),  P,  of  the  jet  against 
the  cups  of  the  wheel,  by  the 
linear  velocity  v  of  these  cups, 
which  is  the  distance  through 
which  the  working  force  acts 
•each  second. 

Fig.  4  shows  such  a  wheel 
in  steady  operation,  supplied 
with  a  free  jet  issuing  from  an  orifice  or  nozzle  in  the  side  of  a 
reservoir  wiiose  upper  surface  is  hi  ft.  above  the  center  of  nozzle. 
The  velocity,  c,  of  the  jet,  since  it  is  a  free  jet,  is  practically  the 
same  as  if  the  wheel  were  not  in  position  and  has  a  value 
(see  §496,  M.  of  E.)  of  c  =  0x/20ST,  where  <£  is  the  co- 
efficient of  velocity  for  the  nozzle  in  question. 

For  uniform  motion  of  the  wheel,  R'  being  the  resistance 
applied  to  the  rim  of  the  smaller  wheel  (on  same  shaft)  where 
the  velocity  is  v',  we  must  have,  from  the  theory  of  work  and 


FIG.  4. 


10  HYDRAULIC   MOTORS.  §  7- 

energy  applied  to  the  uniform  motion  of  this  rigid  body,. 
Pv  =  R'vf.  But  from  p.  808;  eq.  (7),  M.  of  E.,  we  have,  for 

a  series  of  cups,  the  value*  of  P,  viz.;  P  =  —          —,  where 

tJ 

Q,  =  Fc,  is  the  volume  of  water  passing  per  second  from  the: 
nozzle.  (F  =  the  sectional  area  of  the  jet.) 

Hence  the  power  expended  on  R'  ',  R'v',  or  exerted  by  Pr 

is  (after  writing  —  for  v,  for  maximum  power;   see  p.  808,  M.. 

of  E.) 

Or   c2 

L=PV=VI.C    R>v>  ......  (i> 

9     * 

Or,   substituting   from   the   equation   c  =  <j>V2ghi, 

L,  =  R'tf,  =  <PQrhi  .......     (2) 

As  the  action  of  the  water  on  the  cups  is  more  or  less  imperfect,, 
the  usual  power  (R'v'}  obtained  in  practice  is  rarely  more  than 
80  per  cent,  of  this  last  expression.  If  this  imperfection  of 
action  could  be  neglected  and  the  value  of  <£  taken  as  unity, 
with  h\  approximating  to  h  (the  total  vertical  distance  between 
head-  and  tail-  water  surfaces),  the  theoretical  ideal  maximum. 
power  of  the  motor  would  be  Qrh,  ft.-lbs.  per  sec.,  as  in  tiia- 
other  cases  already  instanced. 

As  before,  the  efficiency  would  be 

flV 


Here  we  may  say  that  the  total  power  of  the  mill-site  Qrh 

c2 
(ft.-lbs  per  sec.),  i.e.,  Qr  —  ,  (if  <£  be  unity,)  has  been  converted 

Or  c2 
into  the  kinetic  form  -—.75-  (or,  mass  per  second  X  half-square 

9     * 

of  the  velocity  of  jet)  at  the  point  where  the  water  is  about 
to  act  on  the  motor;  so  that  this  kind  of  motor  utilizes  the 
energy  of  the  mill-site  in  the  kinetic  form.  At  the  point  of 
leaving  the  motor  the  water  is  at  the  same  level  as  at  entrance^ 
and  is  under  the  same  pressure  (atmospheric  pressure)  as  at 

*  This  value  of  P  is  also  proved  in  this  book.     (See  eq.  (6),  p.  66,  with  a 
180°.) 


§  8.  TYPES    OF   HYDRAULIC  MOTORS.  11 

entrance,  but  has  practically  lost  all  its  velocity  (when  cups 
have  proper  speed). 

Numerical  Example. — With  a  head,  hi,  of  100  ft.  and  a 
value  0.95  for  <£>,  we  have  for  the  velocity  of  the  jet  (free 
jet)  c  =  0.95  XV2X  32.2X100,  or  76.23  ft.  per  second.  If 
the  mill-site  furnishes  Q  =  2  cu.  ft.  of  water  per  second,  the 
kinetic  power  (i.e.,  kinetic  energy  per  second)  of  the  jet  just 
before  impinging  on  the  cups  of  the  wheel  is 

Or  &_  _  2X62.5  (76.23)2 
g'2*"       32.2          2       ' 

i.e.,  11280  ft.-lbs.  per  second.  If  the  impulse  wheel  utilizing 
this  jet  has  an  efficiency  of  80  per  cent.,  the  useful  power  ob- 
tained will  be  L',  =  #V,  =  0.80  X 11280  =  9024  f  t.-lbs.  per  second; 
for  which  result  the  speed  of  the  cups  must  be  maintained 
at  the  proper  value,  viz.,  c-^2,  or  38.1  ft.  per  second.  To  keep 
the  speed  of  the  cups  from  accelerating  beyond  this  figure 
the  value  of  the  resistance  Rf,  if  it  is  to  act  on  a  periphery  of 
the  wheel  having  (say)  half  the  radius  of  that  described  by 
the  center  of  the  cups,  will  need  to  be 

Rf  =  L'  +  v'  =  9024  - 19.05  =  473  Ibs.     (The  value  of  P  is  236  Ibs.) 

In  the  case  of  an  impulse  wheel  the  efficiency  is  usually 
referred  to  Qj-hi  instead  of  Qj-h  (see  Fig.  4). 

8.  Mixed  Types  of  Motors. — It  will  be  seen  later  that  in 
the  working  of  some  kinds  of  motors  (like  the  class  termed 
"  reaction-turbines ")  the  water  is  not  only  under  pressure 
in  closed  spaces  at  the  entrance  of  the  motor  channels,  but  may 
have  considerable  velocity  as  well.  In  other  words,  the  energy 
of  the  water  at  entrance  is  partly  in  the  pressure  form  and 
partly  in  the  kinetic.  It  will  therefore  be  of  interest  and 
advantage  to  prove  a  general  theorem  of  such  a  form  as  to 
bring  into  play  all  three  of  the  quantities  pressure,  velocity, 
and  ekvation  (above  a  convenient  datum)  of  the  point  where 
the  water  enters  the  motor,  or  just  before;  and  also  similar 
quantities  at  the  point  of  exit  from  the  motor,  or  just  down- 
stream from  such  a  point,  as  follows: 


FIG.  5. 


12 


§  9.      GENERAL  THEOREM  FOR  HYDRAULIC  MOTORS.      13 

9.  General  Theorem  for  the  Power  Derived  from  any  Hy- 
draulic Motor  in  Steady  Operation. — This  will  apply,  what- 
ever the  nature  of  the  motor  may  be  (piston-motor,  rotary- 
motor,  or  what  not)  so  long  as  its  operation  is  smooth  and 
steady,  with  uniform  motion  of  the  parts  and  a  steady  flow 
on  the  part  of  the  water  at  rate  of  Q  cu.  ft.  per  sec.  Fig.  5 
shows  a  casing  M,  within  which  a  water-motor  is  working. 
Water  enters  at  n  through  a  pipe  AB,  shown  in  longitudinal 
section,  and  leaves  the  motor  at  m  through  the  pipe  EL.  All 
pipes  are  supposed  full  of  water,  as  also  all  chambers,  cells, 
or  passageways  of  the  motor,  which  is  composed  of  rigid 
parts.  Piezometers  Pn  and  Pm  being  supposed  inserted  in 
the  walls  of  the  pipes  at  AC  (up-stream  pipe)  and  EK  (down- 
stream pipe),  the  internal  fluid  pressure  at  point  n,  viz.,  Pn, 
will  be  indicated  by  the  height,  yn,  of  the  stationary  water 
column  (plus  the  atmospheric  pressure,  since  the  piezometer  is 
an  open  one).  That  is,  with  pa  for  atmospheric  (unit)  pressure, 
we  have  pn  =  pa  +  ynj"}  and  likewise  at  the  point  m  the  internal 
fluid  pressure  is  pm  =  pa  +  ymT'  The  mean  velocity  of  the 
water  in  the  cross-section  of  the  pipe  at  n  may  be  called  vn ; 
and  that  at  m,  vm.  The  height  of  n  above  the  datum  plane 
in  figure  will  be  called  zn]  that  of  m,  zm.  Elevation  of  n 
above  m  =  hf'}  and  the  difference  of  elevation  of  the  summits 
of  the  two  piezometer  columns,  h. 

The  power  of  the  motor  is  considered  to  be  applied  to  the 
overcoming  of  a  constant  resistance,  Rf  Ibs.,  in  the  form  of 
the  tension  in  a  rope  or  cable  the  velocity  of  any  point  of  which 
is  constant  and  is  denoted  by  v'.  That  is,  the  cable  is  being 
wound  upon  a  drum  at  a  uniform  rate.  The  value  of  Rf  is 
supposed  to  be  such  that  the  motion  of  the  motor  and  of  the 
water  passing  through  is  "steady,"  so  that  no  part  of  the  motor 
has  any  acceleration  and  the  values  of  pn,  pm,  vn,  and  vmj 
and  also  that  of  Q  (cubic  feet  per  second,  rate  of  flow  of  the 
water)  remain  constant.  The  sectional  areas  of  the  two  pipes 
at  n  and  m  are  Fn  and  Fm,  respectively;  whence  we  have 
Q  =  Fnvn',  and  also  Q  =  Fmvm. 

We  are  now  to  consider  the  assemblage,  or  collection,  of 


14  HYDRAULIC   MOTORS.  §  9. 

rigid  bodied  consisting  of  all  the  particles  of  water  between 
the  sectional  plane  AB  of  the  inlet-pipe  and  that,  EL,  of  the 
outlet-pipe,  together  with  all  the  moving  parts  of  the  motor 
itself  (including  the  cable  up  to  the  point  where  Rf  is  applied 
(see  Fig.  5).  To  the  range  of  motion  of  this  collection  of 
rigid  bodies  which  takes  place  during  a  short  time,  dt  seconds, 
let  us  apply  the  general  Theorem  of  Work  and  Energy  as  proved 
in  §  142,  p.  149,  of  M.  of  E.  In  this  theorem  the  work  done 
by,  or  upon,  those  forces  only  which  are  external  to  the  bodies 
concerned  need  be  considered,  pressures  between  any  two 
bodies  of  the  collection  being  totally  ignored  unless  of  the 
nature  of  friction.  Both  internal  and  external  frictions  must 
be  considered,  each  internal  friction  (i.e.,  friction  between 
any  two  members  of  the  collection)  must  be  multiplied  by 
the  proper  distance  of  relative  travel.  Any  external  force 
whose  point  of  application  moves  at  right  angles  to  the  line 
of  the  force  is  "  neutral/7  i.e.,  does  no  work/ 

Items  of  Work. — During  the  small  time,  dt  seconds,  here 
considered,  the  pressure  on  the  bounding  plane  AB,  viz., 
Fnpn  Ibs.,  is  a  working  force  and  works  through  the  small  dis- 
tance dsn,=BD,  the  work  so  done  being  Fnpndsn,  while  section 
AB  is  moving  to  a  new  position  CD.  Similarly,  the  resisting 
pressure  Fmpm  on  the  down-stream  bounding  plane  (as  if 
it  were  the  face  of  a  piston)  EL,  at  m,  is  overcome  through  a 
-corresponding  small  distance  dsm,=LH;  i.e.,  the  work  Fmpmdsm 
is  spent  upon  this  resistance.  The  resistance  R'  is  overcome 
through  some  small  distance,  dsf  (amount  of  cable  wound  up 
in  time  dt).  The  work  expended  on  external  friction,  such 
as  axle,  or  shaft,  friction,  will  be  indicated  by  2(R"ds"),  while 
that  of  internal  friction  (of  the  water  on  itself  or  between 
the  water  and  the  moving  blades  or  pistons  of  the  motor)  by 
2(R'"d8"f).  During  the  motion  of  this  collection  of  rigid 
bodies  in  time  dt,  the  center  of  gravity  of  the  portion  of  water 
now  being  considered,  situated  initially  between  AB  and  EL, 
the  weight  of  which  we  may  call  G  Ibs.,  sinks  from  some  posi- 
tion a  to  some  lower  position  b.  Denote  the  length  of  the  ver- 
tical projection  of  this  distance  a  . .  b  by  dh  (feet).  This  gravity- 


§  9.     GENERAL  THEOREM  FOR  HYDRAULIC  MOTORS.      15 

force,  G,  does  the  work  G-dh.  When  the  plane  AB  arrives 
at  CD  (and,  correspondingly,  plane  EL  reaches  position  KH) 
there  is  as  much  water,  and  in  the  same  position,  between 
CD  and  EL  as  there  was  before,  and  the  weight  of  the  lamina 
AD  equals  that  of  lamina  EH  (viz.,  FndsnY  =  Fmdsmf)',  hence 
the  product  of  weight  Fndsnr  by  the  vertical  height  hf  (  =  zn—zm) 
is  equal  to  that  of  weight  G  by  dh  (see  §  32  for  more  detailed 
proof)  and  may  replace  it.  Therefore,  finally,  the  expression 
for  the  aggregate  work  done  (positive  and  negative)  in  the 
time  dt  is 

•dW  =  Fnpndsn  -  F  mpmdsm  +  (Fndsnr)  (zn  -  zm)  -  E'ds* 

-2(R"ds")-Z(R"'dsm).     (1) 

It  still  remains  to  formulate  the  change  that  occurs  during 
this  time  dt  in  the  amount  of  kinetic  energy  possessed  by  the 
moving  rigid  bodies  of  the  collection  considered.  Since  the 
motion  of  all  the  parts  of  the  motor  itself  is  uniform,  such 
change  for  them  will  be  zero.  As  for  the  (rigid)  particles  of 
liquid  concerned,  consider  all  the  particles  of  water  between 
AB  and  EL  to  be  divided  into  a  vast  number  of  contiguous 
groups,  of  equal  volumes,  each  group  having  a  volume  equal 
to  that  of  the  lamina  ACDB.  this  lamina  being  the  first  group 
of  the  series  and  having  a  mass  =  dAf,  =  FndsnY+-g;  these  groups 
being  so  selected  that  in  the  short  time  dt  the  velocity  of  all 
the  particles  in  any  group  shall  have  acquired  a  new  value 
just  equal  to  that  which  the  particles  of  the  group  next  ahead 
had  at  the  beginning  of  the  dt.  It  follows,  therefore,  from 
the  definition  of  " steady  flow"  (see  p.  648,  M.  of  E.)  that  in 

/dMv2\ 
subtracting    the  initial    kinetic   energy!  —••—  jfrom  the  final, 

for  each  group  of  particles,  and  adding  up  these  results  for  all 
the  groups,  from  the  first,  AD,  to  the  last,  EH,  (whose  right- 
hand  face  is  at  EL  at  the  beginning  of  the  dt,)  all  the  terms 
involved  will  cancel  out  except  the  initial  kinetic  energy  of  the 
first  group  (or  lamina)  and  the  final  kinetic  energy  of  the  last 
group.  That  is  to  say,  the  result  for  the  aggregate  change  in  the 
Jdnetic  energy  of  all  the  bodies  of  the  collection,  in  time  dt,  is 


16  HYDRAULIC    MOTORS.  §  9~ 

T     Vn2 

'--    ...     (2) 


Equating  the  expressions  for  dTF  and  df(K.E.),  and  re- 
placing Fmdsm  (and  also  its  equal  Fndsn)  by  Q-cft  (the  volume 
flowing  in  time  dt)  and  then  dividing  through  by  dt,  noting 
that  ds'  +  dt  =  v',  the  velocity  of  a  point  in  the  cable,  while  v" 
is  the  velocity  of  the  rubbing  parts  for  any  friction  such  as 
Rr,  and  v"r  has  a  similar  meaning  (relative  velocity)  for  any 
internal  friction,  R'",  we  have,  finally, 


=  #  v  +  1  (#'  V)  +  ^  (#"  V")  .    (3; 

(Each  side  of  this  equation  is  ft.-lbs.  per  second;  power.) 

R'v'  may  be  called  the  useful  power  of  the  motor  and  the 
other  items,  I(R"tf')  and  2(R'"v'"),  the  lost  power,  or  that 
wasted  in  friction.  We  may  therefore  say  that  the  power 
of  the  motor,  partly  spent  in  the  useful  power,  R'v',  and  the 
remainder  wasted  in  the  work  of  friction  (both  of  fluid  friction 
and  that  between  solids)  is  equal  to  the  product  of  the  weight 
Qf  (Ibs.  of  water  used  per  second)  by  the  difference  between 
the  sum  of  the  three  heads  (viz.,  velocity-head,  pressure-head, 
and  potential-head  or  elevation  above  datum)  at  the  point  of 
entrance  to  the  motor,  and  the  sum  of  those  at  the  point  of 
exit  therefrom. 

QT   v2  Or 

Just  as  —  .  —  is  called  the  kinetic  energy  of  the  mass  — 
9     2  g 

of  water,  as  due  to  its  velocity  v,  and  Q?z  its  potential  energy 

7) 

due  to  elevation  above  datum,  similarly  Qf  -  may  be  called 

the  "pressure-energy"  due  to  internal  fluid  pressure;  (a  mere 
name,  however;  useful  when  the  flow  is  steady;  this  would 
not  imply  that  a  receiver  full  of  stationary  water  under  pressure 
possessed  thereby  more  than  a  trifling  amount  of  energy,  due 
to  its  pressure.) 

Hence  eq.  (3)  might  be  reread  as  follows:  The  amount 
of  energy  (of  the  three  kinds  defined)  lost  during  passage 


§  10.    GENERAL  THEOREM  FOR  HYDRAULIC  MOTORS.       17 

through  the  motor  by  the  weight  (say  Ibs.)  of  water  used  per 
second,  is  equal  to  the  power  spent  by  the  motor  on  the  useful 
resistance  and  the  various  frictional  resistances. 

In  actual  practice,  with  a  good  motor  run  at  proper  speed, 
the  useful  power,  R'v'  ',  may  be  as  much  as  85  per  cent,  of  the 
power  given  up  by  the  water;  i.e.,  may  be  85  per  cent,  of  the 
sum  of  the  useful  power  and  the  power  wasted  in  friction  (this 
latter  part  reappears  in  the  form  of  heat)  . 

[N.B.  Evidently,  if  no  motor  is  placed  in  the  line  of  pipe 
between  n  and  m,  R'v'  and  R"v"  disappear  and  we  have  from 
eq.  (3)  Bernoulli's  Theorem  for  steady  flow  in  a  stationary 
rigid  pipe,  the  loss  of  head  between  n  and  m  being  represented 


10.  Another  Form    of   Equation   (3).  —  In  Fig.   5  we  may 
note  the  following  relations  (6  being  the  height  of  the  water 
barometer,  or  about  34  ft.)  : 

T)  T) 

-7T  =  yn  +  b,    and    -?  =  ym  +  b; 
also  hf  +yn  =  ym  +  h,    and    zn-zm  =  h'. 

h  denotes  the  vertical  distance,  or  "drop,"  from  the  summit 
of  the  up-stream  piezometer  column,  at  A,  to  that  in  the  lower, 
at  E.  Eq.  (3)  may  now  be  written  in  the  form 

-).  .    (4) 

Hence,  if  the  entrance-  and  exit-pipes  were  equal  in  sectional 
area,  thus  making  vn  equal  to  vmj  we  should  have 

Qrh  =  R'tf  +  2(R"i/')  +  I(R'"i/").       ...     (5) 

11.  Numerical  Example  of  Foregoing.  —  In  a  test  of  a  hy- 
draulic motor  it  is  found  that  when  a  value  of  R'  (friction  of 
a  brake  on  pulley)  of  240  Ibs.  is  furnished  for  the  motor  tp 
work  against,  on  the  rim  of  a  pulley  of  r=l  ft.  radius  keyed 
upon  the  shaft  of  motor,  the  uniform  speed  to  which  the  motor 
adjusts  itself  is  n  =  306  revs,  per  minute,  the  consumption  of 
water  is  Q  =  l.2  cu.  ft.  per  second,  while  the  pressure-gauge 


18  HYDRAULIC    MOTORS.  §   11. 

readings  at  n  and  m  respectively  (see  Fig.  5)  are  56  Ibs.,  and 
6  Ibs.,  per  sq.  in.,  above  the  atmosphere.  The  point  n  in  the 
supply-pipe,  which  is  4  in.  in  diameter,  is  at  an  elevation  of 
4  ft.  above  the  point  m  in  the  discharge-  or  "'exit  "-pipe,  6  in. 
in  diameter. 

Required  the  useful  power,  R'vf,  and  the  efficiency  of  the 
motor,  )?,  at  this  speed. 

Solution.  —  Here  we  have  (see  Fig.  5),  using  ft.,  lb.;  and 
second, 


Also,      vn  =  Q-*-FB  =  1.20-[j(:j|j      =13.7  ft.  per  sec., 

[/  a  \  2~| 
4\i2/  ]=6-lft-  Persec-; 

Q  Q 

whence  ^-  =  2.91  ft.,     and    ^-  =  0.58  ft. 


Hence 


=  9290ft.-lbs.  per  sec. 

=  energy  given  up  by  the  water  each  second  in  passing  through 
the  motor. 
Now  the  useful  power  being  R'v',  viz., 

W,  =  Rf  (2nrn)  ,  =  240  X  2x  X  1  X  5.  10  =  7690  ft  .-Ibs.  per  sec., 

it  follows  that  in  this  test,  at  speed   of  306  revs,  per  minute, 
the  motor  developed  an  efficiency  of  83  per  cent.  ;  since 

R'vf  7690 


929° 


=  0.83. 


The  difference  between  the  9290  and  the  7690,  i.e.,  1600  ft.- 
Ibs.  per  second,  is,  of  course,  the  value  of  the  lost  power  (heat)  ; 
amounting  to  some  17  per  cent,  of  the  power  given  up  by 
the  water.  At  other  speeds,  to  secure  which  the  value  of 
Rf  would  have  to  be  changed  to  various  other  values,  succes- 


§  12.     GENERAL  THEOREM  FOR  HYDRAULIC  MOTORS.       19 

sively,  the  efficiency  would  be  different;  and  it  is  usually  an 
important  object  in  the  testing  of  a  motor  to  ascertain  at 
what  speed  it  develops  the  greatest  efficiency,  this  speed  being 
the  "'best  speed  "  for  its  operation.  The  quantity  of  water 
used  per  second,  Q,  may,  or  may  not,  be  the  same  at  different 
speeds;  this  depending  on  the  kind  of  motor  employed. 

12.  Pump,  instead  of  Motor.  —  In  this  connection  it  will 
be  of  advantage  to  consider  another  kind  of  test.  In  Fig.  5 
conceive  a  pump  of  some  kind,  say  a  centrifugal  pump,  the 
theory  of  which  will  be  presented  later,  to  be  placed  inside 
of  the  casing  M  and  to  be  operated  by  the  application  of  work- 
ing force  P  Ibs.,  applied  tangentially  to  the  periphery  of  a 
pulley  (radius  =  r)  keyed  upon  the  shaft.  If  the  rim  of  this 
pulley  travels  with  a  velocity  v  and  the  force  P  is  the  tension 
in  an  unwinding  cable  (or  perhaps  the  tangential  component 
of  the  pressure  of  a  pinion-tooth  against  the  tooth  of  a  gear- 
wheel), the  power  applied  in  working  the  pump  will  be  Pv  ft.  -Ibs. 
per  second,  and  water  will  be  caused  to  pass  in  steady  flow 
through  the  pump  from  point  m,  in  what  is  now  an  inlet-pipe, 
to  point  n  in  the  pipe  AB  (now  a  discharge-pipe);  that  is, 
from  a  point  where  the  pressure-head,  velocity-head,  and  poten- 

^  n'j     2i 

tial-head  are  —  ^,  ~  ,  and  zm,  respectively,  to  a  point  n  where 

/  J/ 

the  sum  of  the  corresponding  heads  is  greater  than  at  m.  (vn 
and  vm  now  point  to  left.) 

If  Q  is  the  volume  of  water  pumped  per  second,  it  is  easily 
proved,  by  the  same  method  as  that  just  followed  in  §  9  (con- 
sidering that  in  the  present  case  P  and  Fmpm  are  working 
forces,  and  Fnpn  and  G  resistances),  that 
Pv-Z(R"v")-2(R'"vf") 


or,  more  conveniently,  that 

-~)'\  +  £(R"^+2(le"V").      .     (1) 
Here,  as  before,  Z(R"fv'"}  denotes  the  power  lost  in  fluid 


20  HYDRAULIC    MOTORS.  §   13. 

friction,  whether  in  the  passages  of  the  pump  or  in  portions  of 
the  stationary  pipes  between  m  and  n;  while  2(R"v")  is  the 
power  lost  in  friction  between  the  solid  parts. 

Eq.  (1)  declares  that,  of  the  applied  power  Pv  necessary 
from  some  external  source  (such  as  a  steam-engine  or  water- 
wheel)  to  operate  the  pump  at  a  certain  uniform  speed,  the 

portions   2(R"v"}    and  Z(R'f'v'"}    (ft.-lbs.   per   sec.)  are   lost, 

[Vn2     t<w2-i 
h  +  -^~ — ~-  ,  is  use- 
fully employed  in  pumping  water.     The  efficiency  of  the  pump 
is  the  ratio,  or  fraction, 


j  •     ,      n_m 

useful  power  l    +2g      2ry 

power  applied  Pv 


13.  Example  of  Test  of  Pump.  —  The  following  example 
represents  very  nearly  the  case  of  a  test  of  a  centrifugal  pump 
used  on  a  hydraulic  dredge  on  the  Mississippi  River.  (See 
pp.  136  to  167  of  the  Report  of  the  Mississippi  River  Com- 
mission for  1903.)  Although  reference  is  now  made  to  Fig.  5, 
it  must  be  understood  that  the  direction  of  flow  of  the  water 
is  from  m  toward  n,  and  that  in  the  place  of  a  resistance  R' 
we  now  have  a  working  force  P;  the  direction  of  motion  of  the 
cable  (if  we  conceive  that  to  be  the  manner  of  operating  the 
pump-shaft)  being  the  same  as  that  of  the  force  P. 

From  gauges  inserted  in  the  sides  of  the  entrance-pipe  (or 
"  suction-pipe")  at  m  and  of  the  discharge-  pipe  at  n,  close  to 
the  pump-casing,  and  various  other  measuring  appliances, 
the  following  data  were  obtained: 

pm  =  3     Ibs.  per  sq.  in.  below  the  atmosphere  ; 
pn  =  4.1   "     "       ft     above    " 


V    2 


vm=13     ft.  per  sec.,     hence     ^-  =  2.7  ft.; 
v«  =  13.7"    "     "  "        '—-  =  2.9  ft. 


§  13.     GENERAL  THEOREM  FOR  HYDRAULIC  MOTORS.       21 

The  delivery-point  n  was  (h'  =  )  3  ft.  higher  than  entry 
point  m.  Q  =  8Q.7  cub.  ft.  per  sec. 

The  steam-engine  driving  the  pump  was  found  to  expend 
power  in  so  doing  at  the  rate  (net)  of  280  horse-power.  There- 
fore Pv  =  280X550  -154,000  ft.-lbs.  per  second. 

It  will  be  noted  that  the  pressure  at  m  was  3  Ibs.  per  sq.  in. 
below  atmospheric  pressure;  in  other  words,  the  height  ym 
of  Fig.  5  is  negative.  This  value,  3  Ibs.  per  sq.  in.,  corresponds 
to  a  piezometric  height  of  6.95  ft.  and  hence  the  value  of  h 
(height  from  summit  to  summit  of  piezometer  columns  in 
Fig.  5)  will  be  9.5 +  3 +  6.95,  =  19.45  ft. 

Hence  the  power  expended  in  pumping  water  is 

if  ~  ^f  ]  =  867  X  62'5[19'45  +  2'9  ~  271 
=  105,400  ft.-lbs.  per  second; 

while  the  power  exerted  in  running  the  pump  is,  as  before, 
154,000  ft.-lbs.  per  sec.  Hence,  for  the  efficiency  of  the  pump 
.in  this  trial,  we  find 

3  =  15^000  =  °'685;    °r    68iPercent' 


CHAPTER  II. 
GRAVITY  MOTORS. 

14.  The  Overshot  Water-wheel. — This  form  of  hydraulic 
motor,  with  others  of  the  same  type,  though  now  nearly  obso- 
lete, will  be  given  a  few  pages  in  the  present  work.*  Fig.  6 
(from  Weisbach's  Mechanics)  represents  a  wooden  wheel  of 
this  class,  revolving  in  a  vertical  plane  on  an  axle  in  firm  bear- 
ings. As  seen  from  the  figure  it  consists  of  two  ring-shaped 
shroudings,  or  crowns,  connected  with  the  axle  by  radial 
arms,  and  a  number  of  floats  or  buckets  inserted  between  the 
crowns  and  forming,  with  them  and  a  cylindrical  boarding 
concentric  with  the  axle,  a  series  of  cells.  The  water  is  sup- 
plied from  a  sluice  or  pen-trough  near  the  top  of  the  wheel, 
and  is  regulated  by  a  gate,  falling  in  a  sheet,  or  jet,  into  the 
third  or  fourth  bucket  from  the  summit.  These  wheels  have 
been  constructed  for  falls  of  from  4  to  70  ft.,  sometimes  re- 
ceiving as  much  as  Q  =  50  cub.  ft.  of  water  per  second;  and 
of  from  3  to  50  or  more  horse-power.  With  high  falls  and 
a  large  supply  of  water  it  is  better  to  use  two  or  more  small 
wheels  rather  than  a  single  large  one,  whose  necessarily  great 
weight  would  be  a  disadvantage.  The  fall  is  measured  from  the 
surface  in  the  supply  channel,  or  pen-trough,  to  that  of  the  tail- 
water.  To  lose  the  least  head  possible  the  wheel  should  hang 
just  tangent  to  the  tail- water;  or,  if  the  level  of  the  latter 
is  variable,  high  enough  to  avoid  contact.  In  Fig.  6  H  is 
the  axle,  B  and  C  its  gudgeons;  DMF}  D'M'F',  the  crowns, 
or  shroudings,  made  in  8  to  16  segments  and  from  3  to  5  in. 
thick,  and  fastened  to  each  other  by  cross  tie-bolts  which 

*  The  "  Fitz  Water- Wheel  Co  "  of  Hanover,  Pa  ,  make  the  "I-X-L  Steel 
Overshoot  Water- Wheel ",  up  to  36  ft.  in  diameter.  A  high  efficiency  is 
claimed. 

22 


§  15. 


THEORY    OF    THE   OVERSHOT   WHEEL. 


23 


also  pass  through  the  arms.     The  cog-wheel  E  serves  to  transmit 
the  power  to  the  machinery. 

More  elaborate  wheels  of  this  class  have  been  built  of  iron, 
with  sheet-iron  floats. 


FIG.  6. 


15.  Theory  of  the  Overshot  Wheel. — In  Fig.  6a  we  have  a 
vertical  section  of  a  wheel  of  this  kind,  revolving  counter- 
clockwise with  uniform  motion  and  overcoming  the  resistance 
Rf  at  the  rim  of  the  gear-wheel  A,  The  (uniform)  velocity 
of  the  buckets  should  not  exceed  5  ft.  per  second  for  small 
wheels,  nor  10  ft.  per  sec.  for  high  overshots.  Otherwise  the 
water  would  spill  prematurely  from  the  buckets  and  also  carry 
away  with  it,  unutilized,  too  great  an  amount  of  kinetic  energy. 

Each  bucket  is  filled  on  passing  some  position  E  at  a  con- 
venient angle  Q  with  the  vertical;  the  jet  entering  it  having  a 
velocity  ci=V2ghi,  nearly;  (say  =  0.95\/2^i).  The  height 


24 


HYDRAULIC    MOTORS. 


§  15. 


of  wheel  is  nearly  equal  to  h,  the  fall  or  vertical  "head  "  from 
head-water,  H,  to  tail-water,  T. 

.  If  the  various  dimensions  of  the  wheel,  the  quantity  of 
water  used  per  second  (Q),  the  velocity  of  rotation  of  the 
wheel,  and  the  resistance  Rf  (Ibs.)  tangent  to  the  rim  of  the 
gear-wheel  A,  are  properly  adjusted  to  each  other,  the  motion 
remains  uniform  and  the  work  per  second  done  by  the  pressure 
of  the  water  on  the  walls  of  the  buckets  or  cells  will  (if  we 
neglect  axle  friction)  be  equal  to  that  spent  per  second  (viz., 


R'v'  ft.-lbs.  per  sec.)  on  the  resistance  (vf  being  the  velocity 
of  the  rim  of  wheel  A).  For  example,  Rf  may  be  the  resisting 
pressure  of  the  teeth  of  a  pinion  keyed  on  another  shaft  operat- 
ing the  machinery  of  a  mill.  The  buckets  should  be  of  such 
shape  and  size,  in  connection  with  the  proper  speed,  as  to 
enable  each  cell  to  hold  its  contents  as  long  as  possible  before 
reaching  the  lowest  position.  N  shows  the  point  where  spill- 
ing begins,  and  K  the  position  where  the  cell  has  completely 
emptied  itself.  During  the  filling  of  a  cell  under  the  jet  the 
pressure  against  the  cell  walls  is  greater  (for  equal  amounts 
of  water)  than  it  is  later  when  the  cell  has  passed  the  jet,  since 
the  water  which  first  enters  receives  thereafter  the  impact 


§  17.  THEORY    OF   THE    OVERSHOT   WHEEL.  25 

of  the  jet,  without  being  driven  out  of  the  cell.  The  power 
due  to  this  extra  pressure  is  called  the  power  due  to  impact. 
Its  total  amount  is  much  less  than  that  due  to  the  steady 
action  of  the  weight  of  the  water  after  the  cell  has  passed  the 
jet,  as  will  be.  seen. 

16.  Power  Due  to  the  Weight  of  the  Water.—  E  may  be 
taken  as  the  point  where  the  full  action  of  the  weight  of  the 
water  begins,  just  after  the  mouth  of  the  bucket  has  passed 
the  jet.  Let  ai  denote  the  radius  of  the  "division  circle'7 
(dotted  in  figure)  or  circle  half  way  out  along  the  radial  depth 
of  a  cell;  a  the  radius  of  the  outer  edge  of  cell;  6,  A,  and  X\ 
the  various  angles  marked  in  the  figure.  The  whole  fall,  h, 
or  vertical  distance  between  the  surfaces  of  the  head-  and 
tail-waters,  H  and  T,  may  be  considered  to  be  made  up  of 
four  parts,  viz.,  hi,  serving  to  generate  the  velocity  attained  by 

1  lc2 
the  jet  just  before  entering  a  cell,  i.e.,  0  to  E,  or  hi=-^  —  ; 

y 

a  part,  E  to  N,  or  h%,  =  a\  cos0  +  asin  A;  a  part,  N  to  K,  or 
hs,  =  a  sin  AI  —  a  sin  A;  the  remainder  being  /i4. 

The  power  due  to  the  weight  of  the  water  in  the  buckets 
may  now  be  written  as  the  product  of  Qr  IDS-  per  second  by 
the  height  h%  throughout  which  there  is  no  spilling,  plus  the 
product  of  a  certain  fraction  (  =  dQir)  of  Qf  by  the  height  hs 
throughout  which,  on  account  of  the  progressive  spilling,  the 
average  weight  of  water  in  action  per  sec.  is  dQf.  (On  the 
average,  d  may  be  put  =0.50.)  We  thus  obtain,  for  the  power 
due  to  the  weight  of  the  water, 


.  .  .  (ft.-lbs.  per  sec.).      ...     (1) 

17.  Power  Due  to  Impact.  —  As  already  stated,  the  pressure 
on  the  bottom  of  a  cell,  while  water  from  the  jet  is  entering, 
is  greater  than  the  weight  of  the  amount  of  water  so  far 
entered,  on  account  of  the  impact  of  the  jet,  so  that  the  work 
per  second  done  upon  this  part  of  the  wheel  is  at  a  greater  rate 
than  if  weight  were  the  sole  cause  of  the  pressure  on  the  cell. 

This  extra  pressure,  and  corresponding  power,  due  to  impact 
will  now  be  evaluated;  it  being  borne  in  mind  that  the  particles 


26 


HYDRAULIC    MOTORS. 


17. 


D 


FIG.  7. 


of  water  impinging  on  the  water  already  in  the  cell  do  not  fly 
out  of  the  cell  after  the  impact,  but  join  the  water  already  in 
the  cell  and  move  on  with  it,  and  with  the  same  velocity. 
(See  Fig.  7.)  The  tangential  component  of  the  force  of  impact 

at  the  division  circle,  OA, 
where  the  impact  may  be  con- 
sidered to  take  place,  may  be 
computed  thus:  Let  i>i  =  the 
velocity  of  a  point  of  a  cell  in 
the  circumference  of  the  divi- 
sion circle,  and  c\  the  absolute 
velocity  of  the  particles  of  the 
jet  where  it  intersects  that  circle. 
a\=  included  angle.  During 
each  small  space  of  time  At  a 
small  mass,  Am,  of  liquid  has  its 
velocity  in  the  tangential  direc- 
tion OT  changed  from  c\  cos  ai  to  vi;  i.e.,  its  motion  in  that 

c\  cos  a\  —  r\ 
direction   suffers   a   negative  acceleration  ol   p=  ^      — ; 

therefore  the  retarding  force  must  be 

Am 
Pi  =  mass X ace.  =  Am  .  P=~jr(c±  cos«i  —  t?i), 

which  is  also  equal  and  opposite  to  the  force  in  direction  OT 

with  which  the  mass  Am  presses  the  bucket.     But  ~rr  = 

0 

=  the  mass  of  water  arriving  per  unit  of  time  in  the  bucket. 
Hence  this  force  or  pressure  can  be  written 

P!-— ^ciqosai-vi];  (Ibs.) 

This  is  simply  the  continuous  pressure,  or  working  force, 
acting  on  the  bucket,  due  to  impact.  The  work  done  by  it 
each  second,  i.e.,  the  power  steadily  obtained  from  the  im- 
pact, for  each  bucket  in  turn,  is  obtained  by  multiplying  the 
force  by  the  distance  v\  through  which  it  works  in  each  second 
in  its  own  direction.  Therefore,  so  long  as  a  bucket  receives 


§  18.  OVERSHOT   WHEELS.  27 


water,  the  work  done  upon  it  by  impact  is  at  the  rate  of 
per  unit  of  time;  i.e.,  the  power  due  to  impact 

=  [Qir  +  g][ci  cosai—  v\\v\  ft.-lbs.  per  sec.     .    .     (2) 

for  each  bucket  in  turn.  But  the  portion  of  jet  intercepted 
between  the  edges  of  two  consecutive  buckets  is  free  to  do 
work  on  the  forward  of  the  two  while  other  work  is  being 
done  on  the  hinder  one;  hence,  if  Q  =  the  volume  of  water 
passing  the  pen-trough  H  per  unit  of  time,  the  rate  of  work, 
or  power,  in  the  long  run,  due  to  impact,  is  found  by  replacing 
the  Qi  of  the  last  equation  by  Q;  thus 

Li  =  —  [ci  cosai—  vi]vi (3) 

If  now  we  ma,ke  v\  variable,  we  find  by  Calculus  that  L\  is 
a  maximum  when  VI=%GI  cosai,  and  therefore,  by  substitu- 
tion, 

Ll  max.  =  Qro~  •  i  COs2  ab (4) 

which,  even  for  ai=0,  would  only  =-~- —  — ,  or  only  one  half 

z    g    z 

of  the  kinetic  energy  of  the  water  supply  before  impact;  and 
this  is  the  maximum  effect  of  impact. 

Hence  it  is  an  object  to  use  but  a  small  portion  of  the  total 
fall  to  impart  entrance- velocity  to  the  water.  We  also  note 
that  if  the  entrance-velocity  is  kept  small,  as  above  advised, 
and  if  the  best  effect  of  impact  is  obtained  for  vi  =  Jci  (nearly), 
Vi  itself  must  be  quite  small.  Hence  a  slow  velocity  tends 
to  greater  efficiency  of  the  wheel,  in  this  respect.  There  must 
be  a  limit,  however,  for  a  slow  motion  requires  a  greater  width 
of  wheel  to  accommodate  the  water,  with  a  consequent  increase 
of  weight,  the  axle  friction  occasioned  by  which  would,  beyond 
a  certain  limit,  consume  more  power  than  that  gained  by  the 
slow  motion;  whence  the  limiting  values  of  velocity  mentioned 
in  §  15. 

18.  Total  Power  of  Overshot. — Adding  the  two  items  of 
power  just  obtained,  viz.,  Lg  and  LI,  due  to  gravity  and  im- 
pact, respectively,  we  have,  as  the  total  power,  L, 


HYDRAULIC   MOTORS.  S  19 


I 
J 


(5) 


ft.-lbs.  per  sec.  The  efficiency  is  Ji^L-^-QfTi,  if  axle  friction  be 
neglected. 

19.  Numerical  Example  of  Overshot.  —  Let  the  whole  fall, 
h,  U  64  u.,  ana  let  30  ft.  be  adopted  for  the  diameter  of  tne 
wheel,  with  Q  =  IQ  cub.  ft.  per  second  as  the  available  water 
supply.  Let  the  radius  of  the  division  circle  be  ai  =  14.5  ft. 
and  the  angles  0,  X,  Xi}  and  a\  be  respectively  equal  to  20°, 
46°,  70°,  and  12°;  (see  Figs.  6a  and  7.)  Compute  the  power 
of  the  wheel,  if  3  ft.  be  taken  as  h\,  and  the  value  of  d  as  0.5. 

With  foregoing  data  we  have,  for  the  entrance-velocity 
of  jet,  c1;  =0.95\/20fo,  =  0.95V  64.4X3  =  13.1  ft.  per  sec.; 
whence  v\  should  be  13.1-^-2,  =6.5  ft.  per-  second,  for  best 
effect  of  impact. 

The  fall  with  full  buckets  is  then  found  to  be 

h2,  =  ai  cos  20°  +  a  sin  46°,  =  14.5  X  0.94  +  15  X  0.719, 

=  13.62  +  10.78  =  24.40  ft. 
Also,    h3  =  a(sm  70°  -sin  46°)  =  15(0.940-0.719) 

=  15X0.221  =  3.31  ft. 
We  have,  then,  for  the  total  power,  from  eq.  (5)  of  §  18, 


=  10  X62.5[1.27  +24.40  +  1.65]=  10x62.5x27.32 
-17,075  ft.-lbs.  per  sec.,  =  31.0  H.P. 

If  there  were  no  axle  friction,  nor  resistance  due  to  the 
atmosphere,  this  power  would  be  equal  to  R'tf  (Rf  being  the 
useful  resistance,  at  edge  of  gear-wheel,  and  v'  the  velocity 

of  that  edge).    If  the  radius  of  the  gear-wheel  is  3  ft.,  the  value 

3 
of  vf  is  j^-r  of  vi,  or  ?/  =  L34  ft.  per  sec.     From  J?V  =  17,075, 

we  have  72'  =  17,075  -J-  1.34  =  12,700  Ibs. 

But  the  weight  of  the  wheel  and  the  water  in  the  buckets 
and  R'  itself  (unless  the  latter  pointed  upward,  as  it  would 
do  if  on  the  other  side  of  the  shaft  from  its  position  in  Fig.  6) 


§  19a.  OVERSHOT   WHEELS.  29 

would  occasion  considerable  axle  friction.  For  example,  if 
the  total  pressure  on  the  main  bearings  of  the  shaft  were 
30,000  Ibs.,  and  the  coefficient  of  axle  friction  were  0.10,  the 
total  friction  would  be  Rn  ',  (  =  0.10X30,000,)  =3000  Ibs.  Sup- 
posing the  diameter  of  each  journal  to  be  6  in.,  we  find  that 
its  circumference  would  rub  against  the  bearing  at  a  velocity 
of  (i  *  14.5)6.5,  =0.113  ft.  per  second,  =  v".  Hence  the  product 
#'V',  =  3000X0.113,  =  339  ft.-lbs.  per  second, 

would  be  the  power  lost   through  this  cause.     This  friction 
being  considered,  therefore,  we  put 

L,  =  17,075,  =  R'tf  +  R"tf',     and  obtain 
JBV  =  17,075  -  339  =  16,736  ft.-lbs.  per  sec.  ; 

that  is,  with  tf  =  1.34  ft.  per  sec.,  #'  =  12,500  Ibs. 

As  to  the  efficiency  of  the  overshot  wheel  in  this  example, 
the  full  theoretical  power  of  the  mill-site  being  Qfh=  10X62.5 
X32,  or  20,000  ft.-lbs.  per  sec.,  we  derive  for  the  efficiency, 

°r    ^TV  Per  cent. 


It  is  seen  from  the  above  figures  that  the  lowest  point  of 
the  wheel  hangs  about  0.5  ft%  above  the  surface  of  the  tail- 
water. 

ipa.  Special  Overshots.  —  The  largest  overshot  ever  built  is 
the  Laxey  wheel,  72  ft.  in  diameter,  on  the  Isle  of  Man, 
England;  developing  some  150  horse-power  and  operating 
pumps  for  draining  a  lead-mine  (see  pp.  214  and  219  of  Cassier's 
Magazine  for  July  1894).  The  largest  wheel  of  this  kind  in 
the  United  States  is  at  the  Burden  Iron  Co.'s  works,  Troy, 
N.  Y.  Its  diameter  is  62  ft.  and  width  22  ft.  and  its  weight 
230  tons,  550  H.P.  being  developed.  The  great  "  sand-wheels  " 
of  the  Calumet  and  Hecla  Mining  Co.,  at  their  stamp-mills  in 
Lake  Linden,  Mich.,  are  practically  reversed  overshot  wheels, 
with  buckets  on  the  rim,  by  means  of  which,  driven  by  suitable 
power,  sand  and  water  are  elevated.  The  diameter  of  each 
is  54  ft.,  and  width  11  ft.  (Cassier's  Mag.,  July  1894,  pp.  217 
and  218).  These  wheels,  as  also  the  Burden  wheel  above 


30  HYDRAULIC   MOTORS.  §  20. 

mentioned,  have  rims  supported  by  tension  spokes,  somewhat 
as  bicycle  wheels  are  constructed. 

Space  cannot  be  given  in  the  present  work  for  developing 
formula  and  rules  for  designing  the  form  and  number  of  buckets; 
to  which  end  simple  geometrical  and  mechanical  principles 
apply,  the  main  point  being  to  have  the  cells  only  partly  filled, 
that  spilling  may  occur  as  late  as  possible.  The  parabolic 
path  of  the  jet  issuing  from  the  head-basin,  or  pen-trough, 
must  also  be  considered  in  arranging  for  the  proper  position 
of  the  sheet  or  jet  entering  the  buckets.  For  details  of  this 
kind  the  reader  is  referred  to  Weisbach's  "  Hydraulic  Motors/7 
translated  by  Prof.  Du  Bois. 

Values  of  efficiency  as  high  as  90  per  cent,  have  been  reached 
by  well-designed  overshots,  but  their  construction  has  been  dis- 
continued for  many  years.* 

20.  Breast  or  Middleshot  Wheels. — Wheels  revolving  in  a 
vertical  plane  and  having  buckets  or  floats  receiving  the  water 
at,  or  near,  the  level  of  the  axle  are  called  middleshot  wheels; 
and  if  set  in  a  flume  closely  fitting  the  water-holding  arc,  Breast 
wheels.  The  "apron"  or  surface  of  the  flume  fitting  the  wheel 
should  not  be  more  than  i  to  1  in.  from  the  circumference  of 
the  wheel,  that  but  little  water  may  escape.  Instead  of  buckets, 
simple  radial  floats  are  generally  used,  sometimes  slightly  curved 
backwards  near  the  circumference,  to  diminish  resistance  on 
rising  from  the  tail- water. 

A  large  number  of  floats  is  effective,  not  only  because  the 
loss  of  water  between  wheel  and  apron  is  smaller,  but  because, 
from  the  smaller  interval  between  them,  the  impact  head  is 
smaller  and  the  vertical  distance  through  which  the  water 
acts  by  gravity  is  greater.  Generally  the  outer  distance  be- 
tween two  consecutive  floats  is  made  =  d  =  the  width  of  the 
shroudings,  i.e.,  from  10  to  12  inches. 

It  is  essential  that  middleshot  wheels  should  be  well  "venti- 
lated," that  is,  provision  should  be  made  for  the  passage  of 
the  air  from  the  bucket-space  toward  the  inside  of  the  wheel; 
since  the  water  on  entering  the  wheel  fills  nearly  the  whole 
cross-section  between  the  floats,  thus  preventing  the  ready 

*  A  notable  revival  of  this  type  of  motor  is  seen  in  the  "  Fitz  "  wheel  men- 
tioned in  the  foot-note  of  p.  22. 


§  21.  BREAST   WHEELS.  31 

escape  of  the  air  outwardly.  This  is  all  the  more  necessary 
from  the  fact  that  the  shrouding-space  of  tljese  wheels  is  filled 
to  one  third  or  one  half  of  its  capacity.  Breast  wheels  are  in 
use  on  falls  of  5  to  15  ft., 
and  using  from  5  to  80 
cub.  ft.  per  second. 

The  water  may  be  in-  |_| 
troduced  either  by  means 
of  an  overfall  weir,  or 
through  a  sluice-weir,  with 
gates.  Fig.  8  shows  a 
vertical  section  of  a  breast 
wheel  in  which  the  former 
method  has  been  adopted. 
If  h0  is  the  "head  on  the 
weir/'  or  depth  over  the 

sill,  while  e  is  the  width  of  the  overfall  (same  as  that  of  the 
wheel),  and  Q  the  volume  of  water  used  per  second,  we  have 
(from  p.  683,  M.  of  E.) 

.......     (1) 


whence  the  required  depth  of  the  overfall  (measured  from  the 
surface  of  still  water  3  or  4  feet  back  of  the  weir)  will  be 


The  value  of  the  coefficient  /*  varies  from  about  0.60  for 
a  sharp-edged  sill  at  upper  edge  of  a  vertical  plate,  to  0.80 
or  more  for  a  rounded  sill.  In  the  wheel  in  Fig.  8  the  sill,  A, 
is  adjustable,  to  suit  different  stages  of  water. 

21.  Power  of  Breast  Wheels.  —  As  in  the  case  of  the  over- 
shot, the  work  per  unit  of  time  is  due  partly  to  impact  at  en- 
trance, but  chiefly  to  the  weight  of  the  water  after  entrance. 
The  whole  fall  h  (see  Fig.  8)  may  be  divided  into  two  parts, 
of  which  the  upper,  hi,  is  the  vertical  distance  from  the  surface 
of  the  head-water  to  the  point  of  impact  of  the  water  on  a 
float;  while  Ji2  may  denote  the  remaining  lower  portion.  As 


32  HYDRAULIC    MOTORS.  §  22» 

in  the  case  of  the  overshot,  let  c\  be  the  velocity  [ci  =  0.95V  'l'jh\\ 
of  the  water  just  Before  impinging  on  a  bucket,  at  the  "  division 
circle/'  and  vi  the  velocity  of  a  point  of  the  float  in  the  "divi- 
sion circle  "  (see  §  16).  Also  let  oQf  denote  the  value  of  the 
effective  weight  (per  second)  of  the  water  acting  on  the  floats 
throughout  the  height  h2.  The  angle  between  c\  and  v\  is  a\. 
We  may  therefore,  by  the  same  reasoning  as  in  the  case  of 
the  overshot,  write  the  work  done  upon  the  wheel  per  second 
by  the  action  of  the  water,  both  by  impact  and  by  weight, 


f 
=  Qr\ 


(ci 

(3) 


As  fro  the  value  of  the  ratio  d,  Weisbach  gives  an  example  in 
which  he  makes  application  of  his  method  of  computation  to 
a  wheel  where  the  distance  between  the  apron  and  the  edge 
of  the  floats  is  j-  in.  (§209,.  Vol.  II),  obtaining  £  =  0.93.  In 
other  cases  where  this  distance  is  larger  than  }  in.,  as  with 
wooden  wheels,  d  would  be  smaller,  since  the  volume  of  water 
escaping  between  the  apron  and  float-edges  would  be  propor- 
tionally greater. 

22.  Modern  Breast  Wheels.  —  In  Figs.  9  and  10  (on  p.  33) 
are  shown  two  varieties  of  breast  wheel  as  manufactured  by 
the  firm  of  A.  Wetzig,  Wittenberg,  Germany.     That  in  Fig.  9 
is  constructed  mainly  of  iron;  the  other  of  wood.     The  iron 
wheel  is  intended  for  flows  of  as  much  as  200  cub.  ft.  per  sec. 
and  for  low  heads  of  2  to  6  ft.;  while  the  wooden  wheel  is  to 
be  used  for  heads  of  8  to  25  ft.,  with  flows  of  3  to  35  cub.  ft. 
per  second.     A  fuller  description  will  be  found  in  Engineering 
News  of  Nov.  27,  1902,  -p.  436.     On  the  left  in  Fig.  9  is  seen 
an  '''emergency  gate,"  capable  of  falling  quickly  into  position 
by  the  release  of  a  rope. 

In  the  first  half  of  the  nineteenth  century  breast  wheels 
were  very  common  in  New  England,  but  were  gradually  dis- 
placed by  the  turbine. 

23.  High  Breast  Wheels,  or  Back-pitch  Wheels.  —  This  name 
is  given  to  wheels  with  buckets,  like  the  overshot,  but  receiving 
the  water  between  the  level  of  the  axle  and  the  summit  of  the 


FIG.  10. 


33 


34 


HYDRAULIC    MOTORS. 


§24. 


wheel.  They  are  also  set  in  a  flume  like  ordinary  breast  wheels; 
and  are  peculiarly  well  adapted  to  situations  where  the  surface- 
level  of  the  head-water  is  liable  to  change,  the  gate  being 
adjustable  to  different  heads,  and  heights  of  orifice.  To  pro- 
vide for  the  easy  escape  of  air  from  the  bucket  as  the  water 
enters,  "ventilation"  is  often  resorted  to,  and  is  especially 
necessary  in  the  case  of  these  wheels.  This  was  first  proposed 
by  Fair  bairn,  in  one  instance  furnishing  a  saving  of  3  per  cent. 
of  power,  by  actual  experiment.  See  §  20. 

24.  Efficiency  of  Breast  Wheels. — As  a  result  of  General 


FIG.  11. 

Morin's  experiments  with  two  breast  wheels,  both  with  well- 
fitting  flumes,  the  efficiency,  >?,  was  found  to  be  =0.60  and 
0.70  respectively.  In  general  TJ  ranges  from  0.65  to  0.75  for 
breast  wheels;  and  for  Wesserly  wheels,  as  high  breast  wheels 
are  sometimes  called,  from  0.65  to  0.72.  Some  overshots  have 
been  found  to  give  efficiencies  of  0.80  and  above. 

25.  The  Sagebien  Wheel. — A  peculiar  variety  of  breast 
wheel,  invented  by  Sagebien,  is  shown  in  Fig.  11.  Its  revolu- 
tion must  be  very  slow  on  account  of  tne  shape  of  the  floats 
and  their  position.  Hence  much  intermediate  gearing  is 
rendered  necessary. 

If  the  direction  of  motion  of  the  Sagebien  wheel  is  reversed, 
it  requires  power  from  without  to  drive  it  and  becomes  a  pump, 


$27. 


UNDERSHOT   WHEELS. 


35 


the  power  spent  upon  it  being  used  to  raise  water.  Such  a 
pump-wheel  has  been  constructed  and  set  up  for  purposes  of 
irrigation  on  the  Nile  in  Northern  Egypt.  (See  the  London 
"  Engineer/7  Jan.  1886.) 

26.  Undershot  Wheels. — These   are  almost  entirely  inertia 
motors,  rejecting  the  water  at  about  the  same  level  as  that 
at  which  it  entered.     The  stream,  with  velocity  c=\/2gh  due 
to  the  head,  issues  horizon- 
tally from  under  a  sluice-weir 

into   the    air  and  leaves  the 

floats  with   about   the    same 

velocity   v    as    that    of    the 

extremities     of     the     floats, 

having  impinged  against  them 

in    its    passage    under    the 

wheel.     The  floats  are  radial, 

or    slightly    curved   from    a 

radial  position.    Fig.  12  shows 

an  ordinary  construction.     Evidently  the  whole  power  is  due 

to  impact. 

27.  Power  of  an  Undershot. — Let  Q'  =  the  volume  of  water 
which  actually  suffers  impact  per  unit  of  time.     Let  v  =  velocity 
of  the  middle  of  a  float,  and  c  that  of  the  water  before  striking 
it.     Then,  as  in  §  17,  we  may  write  (remembering  that  a\  is 
zero  here)  for  the  total  power 


FIG    12. 


This  is  a  maximum  for  v  =  %c,  but  even  then  equals  only  half 
the  initial  kinetic  energy  of  the  water,  i.e., 


2    g 


(2) 


Hence,  even  if  Qf  =  Q  (the  volume  of  water  issuing  from 
the  sluice- opening  per  second),  T?  for  undershot  wheels  could 
never  exceed  0.50.  Roughly,  Gerstner  has  computed  from  the 
experiments  of  the  next  paragraph  that  the  power  of  a  good 


36 


HYDRAULIC    MOTORS. 


§28, 


undershot  is  L  = 


(c  —  v)v 


Q-JT,  where  Q  is  the  volume  of  water 


passing   the    sluice-opening   per  time-unit;    and    therefore  Qr 
must  =  about  three  fourths  of  Q. 

As  a  best  velocity  for  wheel  circumference,  Gerstner  gives 
v  =  OAc.  In  general  it  may  be  said  that  the  efficiency  of  under- 
shots  ranges  from  0.25  to  0.33  for  the  ordinary  variety. 

28.  Experiments     with    Undershot    Wheels,    by    Smeaton, 
Bossut,  Morin,  and  others,  have  given  somewhat  varying  results. 
Smeatori,  with  a  small  wheel  75  inches  in  circumference,  found 
T)  no  higher  than  0.25,  while  Bossut,  with  slightly  larger  wheels, 
obtained  a  somewhat  greater  value.     (See  above.) 

29.  Current- wheels  or  Paddle-wheels.  -  -  These    names    are 
given  to  an  undershot  water-wheel,  with  comparatively  few 
radial  blades,  hanging  in  an  open  current  and  supported  on  a 
pier;   or,  more  advantageously,  when  the  height  of  the  water 
surface  is  variable,  upon  a  floating  dock  or  barge.     They  utilize 
less  energy  than  the  common  variety  of  undershot  just  men- 
tioned; not  being  enclosed  in  a 

flume  and  having  fewer  floats. 
Fig.  13  shows  a  simple  construc- 
tion. 

Current-wheels  are  in  use  for 
operating  dredges  on  the  river 
Rhine;  and,  in  a  crude  form, 
have  been  constructed  and  used 
to  some  extent  on  the  streams  of 
the  western  part  of  the  United 
States  for  irrigation  and  other 
purposes.  For  instance,  one  was  constructed  at  Payette  Valley, 
Idaho,  28  ft.  in  diameter,  and  having  28  paddles,  each  16  ft. 
long  and  2.5  ft.  wide.  (See  Engineering  Record,  Nov.  1904, 
p.  621.) 

30.  Poncelet  Undershots.  —  In   this   peculiar   and  efficient 
wheel  the  floats  are  curved  (see  Fig.  14)  and  the  crowns  rather 
deep,  the  wheel  being  so  designed,  and  run  at  such  a  speed, 
that  the  water  enters  without  impact,  mounts  the  curved  side  of 


FIG.  13. 


§30. 


PONCELET   WHEELS. 


37 


a  float  to  a  certain  height  and  then  descends,  exerting  a  con- 
tinuous pressure  and  losing  its  absolute  velocity  gradually; 
and  leaving  the  float  in  a  direction  (relatively  to  the  end  of 
the  float)  opposite  to  that  of  entrance  and  at  the  same  level. 
They  are  specially  suitable  for  small  falls,  under  6  ft.,  utilizing 
about  double  the  energy  of  an  ordinary  undershot.  With 
greater  falls  they  are  excelled  by  breast  wheels  and  are  more 
difficult  of  construction.  The  wheel  must  fit  the  flume  very 
accurately  for  the  best  results.  Poncelet  wheels  have  been 
built  from  10  to  20  ft.  in  diameter  with  32  to  48  floats  of  sheet 
iron  or  wood,  iron  being  the  better  material. 


FIG.  14. 

This  variety  of  undershot  owes  its  superiority  in  efficiency, 
when  compared  with  the  ordinary  undershot,  to  the  fact  that 
the  water  is  received  upon  the  float  at  A  without  impact,  and 
leaves  the  wheel  at  B  with  but  little  absolute  velocity.  At  A 
the  absolute  velocity  of  the  jet  (that  is,  velocity  relatively  to 
the  earth)  is  w=V2gh.  The  edge  of  the  float  has  a  velocity  of 
V  =  B,  little  more  than  one-half  c  and  in  a  different  direction.  If, 
therefore,  a  parallelogram  of  velocities  be  formed  with  w  as 
diagonal  and  v  as  one  side,  the  other  side  c  is  determined  (see 
Fig.  14,  at  A)  and  the  curve  of  the  float  should  be  made  tangent 
to  c,  and  not  to  wt  since  c  is  the  velocity  of  the  entering  water 
relatively  to  the  edge  of  the  moving  float  (see  p.  90,  M.  of  E.),  in 
order  that  the  path  of  the  water  may  suffer  no  sudden  change  of 


38  HYDRAULIC    MOTORS.  §  30a. 

direction.  The  water  reaches  a  height  C  and  then  descends  along 
the  float,  exerting  continually  a  forward  pressure  against  the 
float  (the  stream  being  open  to  the  atmosphere  on  the  other  side). 
On  reaching  the  exit-point,  B,  the  relative  velocity  of  the  water 
is  tangent  to  the  lower  extremity  of  the  float-curve,  as  at  en- 
trance, and  has  about  the  same  value  c  as  at  entrance,  but  is 
now  directed  nearly  backward  as  regards  the  motion  of  the 
wheel.  The  result,  therefore,  of  combining  this  relative  velocity 
with  the  velocity  v  of  the  float-tip  itself  is  to  give  an  absolute 
velocity  of  exit  wn  for  the  water  which  is  small  in  value  and 
nearly  vertical  in  direction. 

Since  at  both  points  A  and  B  the  water  is  under  the 
same  pressure  (atmospheric)  and  these  points  are  practically 
at  the  same  level,  the  energy  given  up  by  the  water  per  second 
is 

Qrw2    Qrwn2 

7^~7^";  '• a 

and  since  impact  is  largely  avoided  (by  means  already  cited) 
a  large  portion  of  this  power  is  transferred  to  the  wheel,  thus 
accounting  for  its  superior  performance.  As  high  an  efficiency 
as  68  per  cent.,  with  v  regulated  to  a  value  of  v  =  0.58w,  has 
been  obtained  by  test.  On  the  whole,  therefore,  Poncelet 
wheels  give  about  double  the  efficiency  of  ordinary  undershots. 
3oa.  Gearing  of  Overshots  and  High  Breast  Wheels. — In 
transmitting  the  power  of  these  wheels,  the  axle  may  be  largely 
relieved  of  the  weight  of  the  water  in  the  buckets  by  so  placing 
the  pinion  which  gears  with  the  cog-wheel  or  rack  concentric 
with  the  axle  of  the  wheel,  that  the  tooth  pressure  between 
the  two  sets  of  teeth  may  be  vertical  and  act  in  the  vertical 
plane  parallel  to  the  axle  and  containing  the  center  of  gravity 
of  the  water  in  the  buckets  at  any  definite  instant.  This 
center  of  gravity  may  be  found  approximately  by  considering 
this  quantity  of  water  as  forming  a  segment  of  a  circular  wire. 
(See  p.  20,  M.  of  E.) 


CHAPTER  III. 

PRELIMINARY  THEOREMS,  FUNDAMENTAL  TO  THE  THEORY  OF 
TURBINES  AND  CENTRIFUGAL  PUMPS. 

31.  Remarks. — Lying  at  the  basis  of  the  theory  of  turbines 
and  centrifugal  pumps  are  the  following  theorems  (A,  B,  and 
C),  the  presentation  of  which  is  necessary  at  this  stage  of  the 
present  work.     The  proofs  of  these  theorems  bring  into  play 
the  fundamental  principles  of  mechanics,  and  it  must  be  .par- 
ticularly noted  that  without  the  existence  of  a  " steady  flow" 
Theorem  C  does  not  hold. 

32.  Theorem  A. — Given  a  homogeneous  mass  abcdefa  whose 
volume  is  V  and  center  of  gravity  at  (7,  Fig.  15;  if  a  thin  hori- 
zontal lamina  AA'  is  removed  from  the  upper  part  and  placed 
so  as  to  occupy  the  space  BB'  (also  in  the  form  of  a  horizontal' 
lamina),  then  the  center  of  gravity  of  the  mass  afb'c'd'defa' 


Ofcv 


PIG.  15. 


(whose  volume  is  also  equal  to  V)  will  occupy  a  position  C" 
at  some  vertical  distance  dH  lower  than  C.  Let  dV  denote 
the  volume  of  the  horizontal  lamina  in  question  and  h  the 

39 


40  HYDRAULIC   MOTORS.  §  33. 

vertical  distance  between  centers  of  gravity  of  AA'  and  BE'  ; 

then  we  are  to  prove  that  V  .  dH  =(dV)  .  h. 

Pass  a  horizontal  reference  plane  OX  through  the  center 
of  gravity  of  lamina  BE'  .  Let  V  denote  the  volume  of  mass 
td'b'cdeja  (mass  common  to  both  arrangements  of  the  com- 
plete mass  of  volume  F).  Then  from  the  properties  of  the 
•"gravity"  coordinates  of  a  mass  and  of  its  component  parts 
.(see  p.  19,  M.  of  E.)  for  the  original  arrangement  of  the  masses, 
•  denoting  by  H'  the  height  of  the  center  of  gravity  of  V  above 
*OX,  we  have 

V.H=V'.H'  +  (dV).h;  .....    (1) 

while  in  the  second  arrangement, 

.....     (2) 


Subtracting  (2)  from  (1)  we  easily  derive 

V  .dH=(dV]  ."h;        Q.E.D  .....     (3) 

33.  Theorem  B.  —  Let  M  be  the  mass  of  a  small  particle 
or  "material  point  "  (Fig.  16)  which  is  describing  a  plane  curve 
ABC  under  the  action  of  one  or  more  forces.  Let  AB  be 
any  element  of  the  path,  described  in  a  time  dt,  while  P  is 
the  resultant  of  all  the  forces  acting  on  the  particle  at  this 
point  of  its  path.  Denote  the  velocity  of  the  particle  in  passing 


A  by  w,  its  velocity  at  B  by  wr,  (each  being  tangent  to  the 
curve  at  its  proper  point,)  while  p  denotes  the  acceleration 
due  to  force  P;  this  acceleration  being  (by  Newton's  law, 
p.  53,  Mech.  of  Eng.)  in  the  direction  of  the  force.  Of  course, 


§  33.  PRELIMINARY    THEOREMS    FOR  TURBINES,    ETC.  41 

If  this  resultant  force  P  were  zero,  the  particle  would  keep 
along  the  straight  line  TD,  tangent  to  curve  at  A  and  the 
velocity  would  remain  constant,  =w.  But  on  account  of  the 
action  of  the  force  P  we  find  that  its  path  curves  away  from 
the  tangent  at  A  and  that  its  velocity  at  B  is  of  a  different 
value,  w'.  If  the  velocity  at  A  had  been  zero,  and  P  had 
then  acted,  the  particle  would  have  moved  in  the  line  of  P 
and  its  velocity  at  the  end  of  dt  seconds  would  have  been  p  .  dt. 
Hence,  by  the  parallelogram  of  motions,  it  follows  that  the 
value  of  w'  must  be  such  as  would  be  given  by  the  diagonal 
of  a  parallelogram  whose  two  sides  are  respectively  equal 
(by  scale),  and  parallel,  to  w  and  to  p  .  dt.  Hence  note  the 
Intersection,  F,  of  the  two  tangent  lines  (at  A  and  B).  A 
parallelogram  whose  side  FD  lies  along  the  tangent  drawn  at 
A  while  the  other  side  is  FR  parallel  to  P  (FD  being  equal 
to  w  and  FR  to  p  .  dt),  must  have  for  its  diagonal  FE,  repre- 
senting wf  in  amount  and  direction. 

Now  the  parallelogram  FE  has  the  same  geometrical  proper- 
ties as  if  it  were  a  parallelogram  of  forces,  that  is,  the  "  moment  " 
of  the  resultant  (diagonal)  about  any  point  is  equal  to  the  (alge- 
braic) sum  of  those  of  its  two  components  (sides)  about  the  same 
point.  Hence,  if  from  any  point,  0,  in  the  plane,  perpendiculars 
are  dropped  upon  the  tangent  line  at  A,  the  tangent  line  at  /?, 
and  the  line  FR,  the  lengths  of  these  perpendiculars  being 
called  k,  k',  and  a  +  n  (n  being  the  perpendicular  distance  of 
P  at  A  from  FR,  while  a  is  the  length  of  the  perpendicular 
dropped  from  0  upon  the  line  of  the  force  P  at  A),  we  may 
write  w'k'  =  wk  +  p  .  dt(a  +  ri),  which  may  be  written 


iv'kf—wk  =  p.dt(a  +  ri),     or    ---  ,//    =a+n,     .     (4) 

since  (w'kf  —  wk)  is  the  increment,  d(wk),  which  the  product 
(wk)  receives  as  a  Consequence  of  the  time  t  taking  an  incre- 
ment dt.  If  now  dt  be  made  equal  to  zero,  the  distance  n 
becomes  zero,  while  d(wk)  +  dt  is  simply  the  "derivative"  or 
first  differential  coefficient  of  the  product  (wk)  with  respect 
to  the  time  t',  so  that,  with  p  =  P  +  M,  we  may  write 


42  HYDRAULIC    MOTORS.  §  34- 

Pa. 

That  is,  the  moment  (Pa)  of  the  force  about  any  point,  =  mass  X 
the  time  rate  of  variation  of  the  product  wk  about  the  same  point.. 

(One  of  Kepler's  laws  for  the  planets  may  be  proved  by 
the  aid  of  this  relation.) 

For  subsequent  use,  this  will  be  written  in  the  form 

M(w'k'-wk)=Pa.dt (5) 

The  quantity  Mwk,  or  Mw'k',  is  called  angular  momentum, 
and  hence  M(wfk'—wk)  may  be  called  the  change  of  angular 
momentum  occurring  in  the  small  time  dt. 

34.  Theorem  C. — Power  of  a  turbine  in  steady  motion  =  an- 
gular velocity  X  change  of  angular  momentum  experienced  by 
the  mass  of  water  flowing  per  unit  of  time,  m  its  passage 
through  the  turbine. — A  turbine  channel  is  essentially  one  of  a 
number  of  short  curved  pipes  or  passageways,  forming  a  single 
rigid  body  (the  turbine,  or  "runner  "),  their  extremities  lying  in 
two  circles  concentric  with  the  axis  of  rotation  (vertical  axis, 
here).  This  set  of  channels  or  "pipes"  (see  Fig.  19)  revolves- 
with  uniform  angular  velocity  (a  sufficient  resistance  being 
offered  to  the  wheel  to  prevent  acceleration,  so  that  the  motion 
is  "'steady  ")  and  water  is  continually  passing  through  them 
and  is  under  pressure;  the  channels,  therefore,  being  always 
full.  The  water  passes  into  these  channels  from  the  mouths 
of  other  and  fixed  (stationary)  passageways  the  walls  of  which 
are  called  guides.  Fig.  18  shows  an  assemblage  of  guides 
which  is  supported  in  the  interior  of  the  ring  (containing  the 
wheel-passages)  of  Fig.  19.  See  also  Fig.  56  on  p.  112. 

Each  vertical  partition  (or  "blade,"  "float,"  or  "vane") 
between  the  wheel-channels  experiences  more  pressure  from 
the  water  on  its  concave  than  on  its  convex  side,  and  the  sum 
of  the  moments  of  all  these  excess  forces,  about  the  wheel 
axis,  called  2 (Pa),  may  be  regarded  as  the  moment  of  a  single 
resultant  couple  representing  the  action  of  the  moving  water 
on  the  wheel,  which  couple  maintains  the  uniform  motion  of 
the  wheel  against  a  proper  resistance.  Suppose  each  force 


GATE 


FALL  RIVER  TURBINE 
Kilburn,  Lincoln^  &  Co 


FIG.  17  (Gate),  1 8  (Guides),  and  19  (Wheel  or  Turbine). 


43 


HYDRAULIC    MOTORS. 


§34. 


tfl 


of  this  couple  to  have  a  value  P0  Ibs.,  with  an  arm  of  a0  ft.; 
then  2 (Pa)  =  P0  .  a0.  In  a  unit  of  time  each  of  the  two  forces 
PQ  works  through  a  distance  wao  +  2,  where  w  is  the  uniform 
angular  velocity  of  the  wheel.  Hence  the  work  per  second, 
or  power  exerted,  is  2P0wa0-^2  =  o>2T(Pa)  ft.-lbs  per  second, 
or  L;  that  is, 

L,  =  power  of  water  on  wheel,  =  ^P0a0  =  col  (Pa) .     .     (6) 

ft.-lbs.  per  sec. 

Now  conceive  the  water  which  at  any  instant  lies  in  the 
turbine-passages  to  be  subdivided  into  a  great  number  of 
vertical  rings,  concentric  with  the  wheel,  of  equal  volumes,  and 

of  such  small  thickness 
that  at  the  end  of  any 
small  time,  dt,  each  ring 
fills  the  exact  space  oc- 
cupied by  its  (forward) 
neighbor  at  the  begin- 
ning of  the  dt.  (See  Fig. 
21.)  The  dotted  line 
MN  in  Fig.  20  shows  the 
absolute  path  (that  is, 
the  path  relatively  to 
the  earth)  and  the  initial  and  final  velocities,  wi  and  wn,  of  a 
particle  of  water,  as  the  ring  to  which  it  belongs  passes  completely 
through  the  wneel. 

(The  curvature  of  this  path  and  the  diminution  of  velocity 
are  to  be  particularly  noted.) 

Fig.  21  shows  the  ideal  division  into  rings  (of  water)  for  a 
segment  of  the  turbine.  In  the  small  time  dt  in  which  any  one 
ring  passes  (outwardly)  into  the  next  consecutive  position,  a 
portion,  A  (of  the  ring),  included  between  any  two  neighboring 
partitions,  or  "vanes,"  passes  into  a  position  A'  in  the  next 
ring-space,  and  in  this  new  position,  on  account  of  the  flow 
being  "steady"  has  an  absolute  velocity  u/,  equal  to  that,  w", 
which  the  portion  B  had  at  the  beginning  of  the  dt]  while  the 
length  of  the  perpendicular,  k',  dropped  on  w'  from  the  wheel 


FIG.  20. 


§34. 


ANGULAR   MOMENTUM        THEOREMS. 


45 


axis,  is  the  same  in  value  as  for  B  at  the  beginning  of  the  dt, 
since  the  positions  A'  and  B  are  in  the  same  ring-space.  In 
other  words,  the  "  moment "  of  the  absolute  velocity  for  Af 
(i.e.,  the  w'k'  of  Fig.  16) 
about,  tne  axis  of  the  wheel, 
at  the  end  of  the  dt,  is  the 
same  in  value  as  that  for 
B  at  the  beginning  of 
the  dt. 

Now  consider  by  itself 
(i.e.,  as  a  "free  body") 
(see  Fig.  22)  the  prism  1,  FIG.  21. 

at  the  entrance  of  any  one  of  the  wheel  channels.  P\  and 
Pi"  are  the  pressures  of  the  partitions  against  it ;  let  P\  repre- 
sent their  resultant;  (it  is,  of  course,  the  equal  and  opposite 
of  the  resultant  pressure  of  the  prism  against  the  wheel  at 
this  instant.)  Since  the  pressures  of  the  neighboring  prisms- 


n 


FIG.  22. 


against  1  have  lines  of  action  containing  0,  the  wheel  axis, 
those  pressures  have  zero  moments  about  0,  and  the  moment 
about  0  of  Pi  (i.e.,  the  moment  P\a\)  is  therefore  equal  to 


46  HYDRAULIC    MOTORS.  §  34. 

the  moment  about  0  of  the  resultant  of  PI,  PI",  and  the 
pressures  on  LH  and  RS.  During  the  time  dt,  prism  1  moves 
to  position  I'  in  the  next  ring-space,  wt  changes  to  w2,  &i  to  k2. 
Hence  from  eq.  (5),  with  dM  as  mass  of  the  elementary  prism, 

dM(wiki  -w2k2)=Piaidt. 

Similarly  for  the  other  prisms  in  this  channel  between  RS  and 
n,  as  they,  simultaneously  with  1,  in  time  dt,  move  into  their 
consecutive  positions,  we  may  write  (remembering  that  all 
the  dM'  s  are  equal) 

dM(w2k2-wsks)=P2a2dt; 
and  so  on,  up  to 


Adding  these  equations,  member  to  member,  we  obtain 

I  (Pa)  for  one  channel  =  -^r(wiki—wnkn). 
Hence,  if  the  wheel  has  m  channels, 

WI//AI 

I  (Pa)  for  the  whole  wheel  =  ~—7r-(wiki—wnkn). 

Now  m  .  dM  is  the  mass  of  water  which  leaves  the  wheel  in 
time  dt;  hence  if  Q  is  the  volume  of  water  passing  per  unit  of 
time,  and  r  is  the  weight  of  a  unit  volume  of  water,  it  follows 
that  mdM  =  (Qr  +  g)  .  dt;  therefore 


Or 
-2*  (Pa)  for  whole  wheel-  —  (wiki  -wjcn),     -     -     (7) 

*J 

which  is  the  moment  of  the  couple  to  which  the  action  of  the 
water  on  the  wheel,  in  this  steady  motion,  is  equivalent.  Hence 
the  power  of  the  wheel  at  this  speed  [that  is,  the  work  per  second 
done  by  this  working  couple]  is,  by  eq.  (6), 

L^w^(Pa)^a^(wlk1-wnkn)    ....     (8) 
y 

ft.-lbs.  per  sec. 

,  The  velocity  wn  =  the  absolute  velocity  of  the  water  at  the 
exit-point  of  a  channel  (see  Fig.  20). 


34. 


ANGULAR   MOMENTUM        THEOREMS. 


47 


The  right-hand  member  of  eq.  (8)  is  seen  to  consist  of  the 
product  of  the  (uniform)  angular  velocity,  w,  by  the  difference 

Or  Or 

between  the  quantities  — wik\  and  — wnkn,  or  the  change  in 


Qr 
of  the  mass,  — ,  of  water  flowing  in 

c/ 


the  "  angular  momentum : 

a  unit  of  time.     (Q.E.D.) 

Eq.  (8)  may  be  thrown  into  a  more  convenient  form,  thus, 
Fig.  23.  By  means  of  a  rectangle,  the  velocity  w\  of  the  water 
at  the  entrance,  M,  of  a  wheel- 
channel  can  be  resolved  into 
two  components,  one,  u\,  tan- 
gent to  the  inner  circle  of  the 
wheel-ring  of  radius  r\,  and  the 
other  along  the  radius  drawn 
to  that  point,  V\. 

Similarly,  at  the  exit-point, 
Nj  of  a  wheel-channel,  the  abso- 
lute velocity  wn  can  be  decom- 
posed into  a  tangential  compo- 
nent un  and  a  radial  component 
Vn,  at  right  angles  to  each 
other.  If  a  and  /*  denote  the 
angles  that  the  absolute  veloci- 
ties make  with  their  respective 
tangents  (Fig.  23),  we  have 

Ui  =  Wi  cosa,     and    un  =  wncos  //. 
Evidently,  from  the  similar  triangles  involved,  we  have 

k\ir\\iUi:w\i     and    kn:rn::un:wn; 
and  hence  eq.  (8)  may  be  written  in  the  form 

wQr 


FIG.  23. 


Power =L  = (utfi  —  unrn) 


'9) 


Hence  the  moment  of  the  couple  would  be 


48  HYDRAULIC   MOTORS.  §  35. 

Again,  since  the  product,  angular  velocity  X  radius,  gives  the 
linear  velocity  of  the  outer  end  of  that  radius,  asri  may  be  re- 
placed by  vi,  the  velocity  of  the  inner  edge  (or  entrance-point) 
of  the  wheel-ring,  and  curn  by  vn,  the  velocity  of  the  outer  edge 
(or  exit-point)  of  the  wheel-ring;  whence  we  may  also  write 

Qr 
Power  of  water  on  turbine  =  L  =  —  (uiVi  —  unv „)      .     (10) 

t/ 

ft.-lbs.  per  sec. 

These  tangential  velocity-components  of  the  water,  u\  and  uny 
are  sometimes  called  the  velocities  of  whirl  of  the  water;  at 
entrance  and  exit,  respectively. 

This  equation  is  remarkable  in  not  involving  the  internal 
fluid  pressures  at  entrance  and  exit. 

35.  Turbine  Pump. — In  the  foregoing  it  has  been  supposed 
that  the  rigid  body  containing  the  set  of  rotating  channels  or 
pipes  forms  a  "turbine,"  the  action  of  the  water  on  which  is 
equivalent  to  a  couple  so  directed  as  to  tend  to  accelerate  the  ro- 
tary motion  of  the  turbine;  which  acceleration  is  supposed  to  be 
prevented  by  application  to  the  turbine  of  a  system  of  re- 
sisting forces  constituting  a  couple  having  a  moment  equal 
and  opposite  to  that  of  the  first,  i.e.,  opposite  in  direction  to 
the  rotary  motion  of  the  rigid  body.     If  the  moment  of  this  first 
equivalent  couple  is  negative  in  any  particular  instance,  it  simply 
shows  that  the  action  of  that  couple  tends  to  retard  the  motion 
of  the  rigid  body,  or  set  of  channels;   for  the  maintenance  of 
whose   uniform   motion,    therefore,    the   second,    equilibratingr 
couple  to  be  applied  to  the  rigid  body  must  have  a  moment 
coinciding  in  direction  with  that  of  tho  rotary  motion  of  the 
rigid  body  itself,  which  in  this  case  acts  as  a  "  centrifugal  pump  "' 
(to  be  treated  in  a  later  chapter;  see  §  105). 

36.  Other  Kinds  of  Turbines. — Although    the    turbine   con- 
sidered in  the  presont  discussion  is  for  simplicity  one  in  which 
the  general  course  of  the  water  is  radially  outward,  in  planes 
at  right  angles  to. the  shaft  of  the  turbine,  the  same  kind  of 
treatment  may  be  applied  whatever  the  nature  of  the  turbine 
in  question  and  corresponding  path  of  the  water  (e.g.,  radial 
inward  flow;   radial  and  downward  flow;   or  one  in  which  the 


§  37.          GENERAL  FORMULA  FOR  TURBINES.  49 

path  of  a  particle  lies  in  a  cylindrical  surface  parallel  to  the 
shaft;  see  later  chapters).  For  any  variety  of  turbine  eq.  (10) 
holds;  vi  and  vn  being  the  respective  velocities  of  the  entrance- 
and  exit-rims  of  the  wheel,  and  u\  and  un  the  projections,  upon 
the  tangent  lines  of  those  rims,  of  the  absolute  velocities  Wi 
and  wn  of  the  water,  at  entrance  and  exit  respectively. 

37.  Numerical   Example. — A   turbine    uses   (2  =50  cub.   ft. 
of  water  per  second  in  steady  operation.     The  absolute  velocity 
of  the  water  at  entrance  is  Wi  =  50  ft.  per  second  at  an  angle 
of  a  =  20°  with  the  tangent  to  wheel-rim ;    and  that  at  exit 
is  ivn  =  W  ft.  per  second  at  an  angle  of  /*  =  110°.     The  speed  of 
the  wheel  is  120  revs,  per  minute,  the  two  radii  being  r\  =  1.5  ft. 
and  rn=2  ft.     Compute  the  power  derived  by  wheel  from  the 
water  under  these  conditions. 

Solution. — From  these  data  we  have,  for  the  "velocities 
of  whirl," 

ui  =50  cos  20°  =50X0.940  =47.0  ft.  per  sec., 
and 

ttn  =  10cos  110°  =  10X(-0.342)  =  -3.42ft.  per  sec. 

Since  the  angular  velocity  =2?r-V2i?  =  12.56  radians  per  sec., 
=  a>,  we  have,  for  the  velocity  of  the  wheel-rim  at  entrance, 
vi  =cori  =18.84  ft.  per  sec.;  and,  for  that  of  outer  wheel-rim, 
vn  =  cur  n  =  25.12  ft.  per  sec.  Hence  the  power  derived  is,  from 
eq.  (10), 

50  X  62  5 
L  =  — 3^H47-°  X  18.84-  ( -3.42)  (25.12)]  =97.1[885.0  +  85.9] 

=  94,300  ft.-lbs.  per  sec.,     or     171.3  H.P. 

We  also  find  that  the  moment  of  the  couple  to  which  the 
action  of  the  water  on  the  turbine  is  equivalent  is 

2 (Pa),  =  L  +w  =94,300  -12.56=7510  ft.-lbs. 

38.  Turbines.     Fundamental  Formula  for  Power. — In    Fig. 

24  is  shown  a  vertical  section  (pulley  in  perspective,  however) 
mainly  diagrammatic,  of  a  (radial  outward-flow)  turbine,  with 
shaft  vertical,  in  steady  operation..  A  is  the  upper  level  or 
head-water,  B  the  lower  level  or  tail-water,  the  difference  of 


FIG.  24. 


§  38.  GENERAL    FORMULA   FOR   POWER.      TURBINES.  5.1 

elevation,  h,  of  their  surfaces  being  the  "head"  of  the  mil) 
site.  The  thick  heavy  lines  indicate  the  turbine  and  its  shaft, 
points  1  being  at  the  entrance  and  n  at  the  exit-point  of  a 
wheel-channel.  In  this  case  the  heights  of  the  wheel-channels 
at  1  and  n  are  not  the  same.  The  guides  are  in  the  space  S, 
S;  the  water  being  conducted  to  them  through  the  rigid  "  pen- 
stock" PP. 

Points  n,  where  the  water  leaves  the  wheel,  are  in  this  figure 
(and  quite  often  in  practice)  at  a  higher  level  than  the  surface 
B  of  the  tail-water;  the  stationary  tubes  or  vessel  which  the 
water  enters  on  leaving  the  wheel  at  n  being  called  the  "draft- 
tube,"  or  "suction-tube."  The  height  hn  is  rarely  taken  at 
more  than  20  ft.,  in  order  that  the  water  in  the  draft-tube  may 
be  under  sufficient  pressure  to  keep  it  full  at  the  highest  point 
n.  In  an  ideal  design  for  the  best  effect  (but  rarely  met  with)* 
the  entrance  of  the  draft-tube  should  be  made  with  a  very 
gradually  enlarging  section  n  to  m,  in  order  to  reduce  to  a 
minimum  the  loss  of  head  at  this  point  in  the  progress  of  the 
water  (more  gradual  than  in  this  figure). 

But  little  leakage  is  supposed  to  take  place  at  1  or  n 
between  the  edges  of  the  moving  crown-plates  (or  shells)  of 
the  wheel  (E,  D)  and  the  stationary  edges  of  the  guides,  or 
of  draft-tubes.  As  already  shown  in  Fig.  19,  the  curved  passage- 
ways or  channels  of  the  turbine  lie  in  a  ring,  being  separated 
from  each  other  by  vertical  curved  blades  or  vanes,  and  are 
closed  in  at  top  and  bottom  by  the  "crown-plates,"  or  shells, 
E  and  D,  which  are  rings,  more  or  less  flat,  providing  a  floor 
and  a  roof  for  each  passageway. 

In  this  figure  the  power  of  the  wheel  is  employed  in  winding 
up  a  cable  on  a  drum  W  keyed  upon  the  shaft  of  the  wheel, 
the  tension  in  the  cable  being  R'lbs.  (Radius  of  drum=r.) 

We  shall  now  assume  that  the  flow  of  water  from  A  to  B 
through  the  fixed  pen-trough,  moving  wheel,  and  fixed  draft- 
tube  is  steady,  and  that  this  flow  takes  place  with  full  passage- 
ways (any  air  previously  contained  therein  having  been  ex- 

*See  Engineering   News,  Dec.  1903,  p.  569. 


52  HYDRAULIC    MOTORS.  §  38. 

pelled),  the  angular  velocity  of  the  wheel  being  uniform,  its 
acceleration  being  prevented  by  a  proper  value  of  R' ',  the 
tension  in  the  cable. 

With  proper  design  of  the  wheel-passages,  etc.,  the  action 
of  the  water  on  the  wheel  is  equivalent  to  a  "  couple  "  acting 
in.  a  plane  at  right  angles  to  the  shaft  and  having  a  certain 
moment  PO«O  ft.-lbs.  The  value  of  this  moment  depends  on 
the  speed  at  which  the  wheel  is  permitted  to  run.  By  "  steady 
motion"  then,  both  of  water  and  wheel,  we  imply  that  the 
resistance  provided  (Rr  Ibs.)  is  such  that  the  moment  R'r  =  P0aof 
where  PO«O  has  the  special  value  corresponding  to  the  par- 
ticular uniform  speed  of  -wheel;  so  that  no  acceleration  takes 
place. 

We  also  assume  that  the  surfaces  A  and  B  are  so  large  that 
the  water  in  these  surfaces  has  no  appreciable  velocity  during 
the  flow;  and  that  all  quantities  concerned  in  the  design  are 
properly  adjusted  to  each  other  to  secure  the  most  advanta- 
geous result  for  the  permissible  consumption  of  water  (or 
rate  of  flow)  Q  cub.  ft.  per  second.  In  other  words,  friction 
is  reduced  to  a  minimum.  This  latter  is  accomplished  by 
such  design  (details  given  later)  that  all  elbows,  sudden  en- 
largements of  section,  eddyings,  etc.,  in  the  flow  of  the  water, 
giving  rise  to  fluid  friction  and  consequent  "loss  of  head"  are 
avoided  (as  much  as  possible). 

Such  being  the  assumptions  made  for  the  wheel  in  Fig.  24 , 
let  us  apply  to  it  and  the  moving  water  the  Principle  of  Work 
and  (Kinetic)  Energy  (see  p.  149,  Mech.  of  Eng.).  This  holds 
good  for  any  collection  of  rigid  bodies  moving  among  each  other. 
The  assemblage  of  rigid  bodies  to  which  we  are  now  to  apply 
it  consists:  first,  of  the  wheel  itself,  with  shaft  and  drum  and 
the  portion  of  cable  shown  in  figure;  the  other  rigid  bodies 
being  all  the  particles  of  water  in  the  whole  body  of  that  liquid 
in  the  two  ponds  and  all  the  internal  spaces  of  the  wheel,  pen- 
stock, and  draft-tube.  (Water  being  practically  incompressible, 
each  particle  of  it  is  a  "rigid  body.") 

The  extent  of  motion  that  is  to  be  considered  is  that  taking 
place  in  a  single  element  of  time,  dt,  seconds.  Since  Q  cub.  ft. 


§  38.  GENERAL   FORMULA    FOR   POWER.      TURBINES.  53 

per  second  is  the  rate  of  flow,  the  quantity  flowing  in  a  time 
dt  will  be  Q  .  dt  cub.  ft.  In  this  short  time,  dt,  surface  A  sinks 
to  Af,  while  surface  B  rises  to  B' ;  the  volume  of  the  hori- 
zontal lamina  of  water  AAf  being  equal  to  that  of  the  lamina 
BB',  each  being  =  Q .  dt.  The  total  atmospheric  pressure 
acting  on  the  surface  A  is  an  external  force  PA,  acting  on  our 
system  of  rigid  bodies,  and  is  a  working  force  doing  the  work 
PAXAA',  while  that  acting  on  Bt  PBy  is  an  external  resistance, 
upon  which  is  expended  the  work  PBXBB'.  Now  these 
products  are  equal  and  cancel  each  other  in  the  summation 
of  items  of  work  (easily  proved  by  the  student). 

In  dt  seconds  the  center  of  gravity  C  of  the  whole  body 
of  water  (whose  weight  we  denote  by  G  Ibs.)  sinks  through  a 
small  vertical  distance  dH.  Hence  the  work  done  by  .this  work- 
ing force  is  G  .  dH  ft. -Ibs;  which,  however,  from  §  32,  eq.  (3), 
can  be  replaced  by  Q  .  dtfh.  Also,  in  this  time  dt,  the  re- 
sistance R'  in  the  cable  is  overcome  a  small  distance,  ab,  or 
ds,  and  the  work  done  upon  it  is  Rf  .  ds.  Disregarding  friction 
for  the  present,  we  note  that  all  the  other  external  forces  acting 
on  the  wheel  and  the  water  particles  (that  is,  the  pressures 
from  the  walls  of  penstock  and  draft-tube  and  the  weight  of 
the  wheel  itself)  are  "neutral"  (that  is,  either  they  act  at  right 
angles  to  the  path  of  the  point  of  application  or  the  point 
of  application  does  not  move  at  all) ;  and  that  all  the  mutual 
pressures  are  normal  to  the  rubbing  surfaces  and  hence  can 
be  omitted  from  consideration.  (See  p.  149,  Mech.  of  Eng.) 

Next,  as  to  the  gain  or  loss  of  kinetic  energy  possessed  by 
each  of  the  rigid  bodies  of  the  collection  considered,  occurring 
between  the  beginning  and  the  end  of  this  small  time  dt;  we 
proceed  thus: 

Conceive  the  whole  body  of  flowing  water  to  consist  of  a 
vast  number  of  very  small  groups  of  particles,  these  groups 
containing  equal  masses  of  liquid  and  so  situated  .  that  in  dt 
seconds  any  group  moves  into  the  position  just  vacated  by 
the  adjacent  group  next  ahead  of  it.  (This  means  that  the 
volume  of  each  group  is  Qdt.  The  lamina  at  A  is  Group 
No.  1,  while  that  at  B  is  the  last  group  of  the  series.)  As  a 


54  HYDRAULIC    MOTORS.  §  38. 

consequence  of  the  flow  being  "steady  "  it  follows  (by  defini- 
tion of  steady  flow)  that  the  particles  of  any  group  acquire  at 
the  end  of  dt  seconds  a  velocity  equal  to  that  which  the  particles 
of  the  next  group  anead  possessed  at  the  beginning  of  the  time 
dt.  Now  in  the  application  of  the  principle  of  Work  and  Energy 
we  are  called  upon  to  subtract  the  initial  amount  of  Kinetic 
Energy  of  each  mass  from  the  final;  but  from  the  circum- 
stances noted  above  it  is  seen  that  the  initial  kinetic  energy  of 
each  group  of  particles  is  equal  to  the  final  kinetic  energy  of 
the  group  just  behind  it,  so  that  when  the  subtractions  indi- 
cated are  all  written  out  and  added  together  a  total  cancellation. 
or  zero,  is  the  net  result  of  the  aggregate  change  of  kinetic 
energy  of  all  the  particles  of  water.  As  already  postulated, 
the  velocity  of  the  group,  or  lamina,  at  A  (and  also  at  B)  is 
insensible.  Also,  since  the  velocity  of  the  rotating  wheel  is 
uniform,  there  is  no  change  of  kinetic  energy  on  the  part  of 
that  body,  in  the  time  dt. 

Hence  the  net  result  of  the  whole  operation  of  applying 
the  method  of  Work  and  Energy  to  the  rigid  bodies  men- 
tioned, for  the  duration  dt  seconds,  is  simply 

G.dH  =  R'.ds,      ......     (1) 

i.e.,  Qrdt.h  =  R'.ds  .......     (2) 

But  (2)  may  be  written 


and  again,  since  ds  +  dt  is  tne  uniform  velocity  of  a  point  in 
the  cable,  or  in  the  rim  of  the  drum  W,  (call  it  v',) 

Qrh  =  R'v'  ........     (4) 

Now  R'tf  is  Ibs.  Xft.  per  sec.,  or  ft.-lbs.  per  sec.,  i.e.,  the  power 
or  rate  at  which  work  is  expended  on  the  resistance  #'(lbs.); 
to  the  steady  and  continual  overcoming  of  which  the  whole 
power  Qj-h,  due  to  the  water  supply  Q  and  the  head  h,  is  ap- 
plied. That  is,  in  this  ideal  case  of  a  water  motor  of  perfect 
design  and  of  perfect  adjustment  in  operation,  with  no  friction 

flV 

of  any  kind,  the  efficiency  =77-7-  =1.00,  or  100  per  cent.;  since 


§  40.  GENERAL  FORMULA.      POWER    OF    TURBINES.  55 

in  general  the  efficiency = ratio  of  the  part  of  the  power  that 
is  usefully  applied,  to  the  whole  theoretical  power  (Qj-h)  of 
the  mill-site, 

39.  Example. — If  the  full  water-supply  is   Q=48  cub.  ft. 
per  second,  and  A  =  100  ft.,  we  have  Qr  =  48X62.5  =3000  Ibsr, 
per  second;    so  that  Qfh  =  300,000  ft.-lbs.  per  second  is  the 
full   theoretical   power   of    the    mill-site    (if    48   cub.   ft.   per 
second  is  the  maximum  available  rate  of  supply).     With  a, 
100  per  cent,  motor  to  utilize    this    power  we  should    have- 
R'vf  =300,000  ft.-lbs.  per  sec.     The  wheel  being  run  at  its  best 
(most  advantageous)  speed  of  (say)  120  revs,  per  minute,  while 
the  radius  of  the  drum  is  r  =  1.5  ft.,  the  velocity  of  a  point  in 
the  cable  would  be  v'  =2nX  1.5x120  ^60  =  18.85  ft.  per  second, 
and  we  have  R'X  18.85  =  300, 000;  i.e.,  R'  is  15,916  lbs.,=the 
tension   that  can  be  overcome  in   the  cable  at  the  specified 
linear  velocity  of  cable  (18.85  ft.  per  sec.). 

Even  with  the  best  designs,  however,  the  useful  power 
obtained  would  rarely  be  more  than  85  per  cent,  of  Qfh  on 
account  of  fluid  friction  and  the  friction  at  the  axle  of  the 
shaft ;  in  such  a  case,  therefore,  we  should  have 

JBV  =0.85X300,000;    [  =  463  H.P;]. 

and  with  v'  the  same  as  before  we  find  that  R'  is  only  13,528  Ibs.. 
(tension,  or  "load  "). 

In  case  the  power  is  taken  off,  not  by  a  cable,  but  by  a 
cog-wheel  gearing  with  a  pinion  keyed  on  the  shaft  of  the 
turbine,  R'  would  represent  the  tangential  component  cf  the 
pressure  between  two  engaging  teeth,  and  vf  the  linear  velocity 
of  the  pitch-circle  of  the  pinion  (or  cog-wheel). 

40.  Turbine    with    Friction. — Referring   again    to    Fig.    24, 
we  note  that  if  both  fluid  friction  and  axle  friction  are  to  be 
considered,  the  outcome  will  be  as  follows: 

Let  h'  denote  the  "loss  of  head  "  that  occurs  between  the 
surface  A  and  the  point  1  where  the  water  enters  the  wheel; 
h"  the  loss  of  head  occurring  in  a  wheel-channel  (that  is,  be- 
tween point  1  and  point  n) ;  and  again  let  h"'  denote  that  occur- 
ring in  the  draft-tube  (that  is,  between  point  n  and  the  sur- 


56  HYDRAULIC    MOTORS.  §  41. 

face  B  of  the  tail-  water).  If  these  three  heads  be  deducted 
from  the  head  h,  the  product  Qr(h  —  h'—h"—h'")  will  express 
the  power  of  the  mill-site  after  fluid  friction  has  been  allowed 
for. 

As  to  axle  friction,  if  that  be  represented  by  R"  Ibs.,  and 
the  uniform  velocity  of  the  circumference  of  the  axle  is  v"  ft. 
per  sec.  then  the  power  lost  in  axle  friction  is  R"v"  ft.-lbs. 
per  sec.;  and  finally 

Qr(h-h'-h"-h'")=R'i/+R"i",  .  .  .  (4a> 
as  applicable  to  a.  turbine  when  friction  is  considered.  If  the 
wheel  runs  immersed  in  water,  another  term,  ROVQ,  might  be 
added  on  the  right  to  represent  the  power  lost  in  fluid  friction 
on  the  wheel-casing  (i.e.,  on  the  outside  surface  of  wheel). 
In  fact,  eq.  (4a)  might  be  written  in  the  form 

Qrh  =R'v'  +R"v"  +  2(R"'vf")  ; 
(see  §  9,  eq.  (3))  the  detail  of  the  term  2(R'"v'")  being 


41.  Bernoulli's  Theorem  for  a  (Uniformly)  Rotating  Channel 
(Steady  Flow  of  Water  Therein).—  See  Fig.  24.  Since  the  steady 
flow  of  water  from  A  to  point  1  occurs  in  a  stationary  (rigid) 
pipe  or  casing,  we  may  apply  Bernoulli's  theorem  for  such 
flow,  denoting  by  p\  the  internal  fluid  pressure  at  point  1, 
by  b  the  height  of  the  water  barometer,  by  7-  the  heaviness 
of  water,  and  by  w\  the  absolute  velocity  of  the  water  at  1; 
whence 

—  -f-^-—  =  b  +  hi  (without  friction).    ...       (5) 

/  */ 

Considering  friction,  we  have 


(&) 


where  h'  is  the  loss  of  head  between  A  and  point  1. 

We  also  note  that  between  points  n  and  B  the  steady  flow 
takes  place  in  a  stationary  (rigid)  pipe,  to  which  if  Bernoulli's 
Theorem  is  applied  (first,  without  friction),  we  have 


§41. 


BERNOULLI  S    THEOREM.      ROTATING    PIPE. 


57 


(6) 


wn  being  the  absolute  velocity  of  the  water  as  it  leaves  the  wheel 
at  n,  and  pn  the  internal  fluid  pressure  at  n;  hn  is  the  height 
of  n  above  tail-water  surface  at  B. 

Usually  there  is  considerable  loss  of  head  between  n  and 
B,  due  to  failure  to  make  the  change  of  section  between  n 


71 


FIG.  25. 


and  m  very  gradual  (Fig.  24).  Calling  this  loss  of  head  h'" 
[as  before  in  eq.  (4a)]  and  applying  Bernoulli's  Theorem  with 
friction,  (n  to  B,)  we  have 


(6a) 


Now  consider,  in  Fig.  25,  the  absolute  path  of  a  particle  of 
water  through  the  wheel;  point  1  is  entrance  where  the  abso- 
lute velocity  is  w\  and  internal  pressure  p\;  while  at  exit, 
N,  wn  is  the  absolute  velocity  and  pn  is  the  internal  fluid 
pressure. 


58  HYDRAULIC    MOTORS.  §  41. 

Since  the  wheel-channel  is  in  motion,  w\  is  not  the  velocity 
of  the  water  at  1  relatively  to  thatj  point  of  the  channel.  This 
relative  velocity  must  be  found  by  drawing  a  parallelogram 
(see  p.  89,  Mech.  of  Eng.)  of  which  the  diagonal  is  made  equal 
to  wi  and  one  side=^i,  the  velocity  of  that  point  of  the  wheel 
(inner  rim),  the  angle  between  these  being  called  a. 

The  other  side,  ci,  being  thus  constructed,  is  the  velocity 
of  the  water  at  1  relatively  to  that  point  of  the  wheel  ("relative 
velocity"  Ci);  and  the  tangent  of  the  wheel-blade  is  made  to 
coincide  with  this,  ci.  in  order  that  the  water  may  follow  the 
blade  at  once  without  having  to  make  a  sudden  turn  or  elbow 
(at  1). 

Similarly,  at  the  exit,  or  point  N,  the  absolute  velocity  wn 
of  the  water  particle  is  the  diagonal  of  a  parallelogram  of  which 
one  side  is  vn,  the  velocity  of  the  outer  rim  of  the  wheel  (which 
is  >vi  in  ratio  of  the  radii  rn  and  ri),  while  the  other  side  is 
Cn,  the  velocity  of  the  water  particle  relatively  to  the  point  n 
of  the  wheel-channel  ("relative  velocity  at  exit").  Of  course 
cn  is  tangent  to  the  extremity  of  the  blade  or  vane  at  point  N. 
Let  d  and  /JL  denote  the  angles  marked  in  Fig.  25  at  point  N. 
Then,  from  trigonometry, 

Ci2  =  Wi2+vi2  —  2viWi  cos  a,   .....     (7) 
and 

cn2=wn2+vn2  —  2vnwncos  fi;  .....     (8) 
hence,  by  subtraction, 

cn2-ci2  =  (wn2-wi2)  +  (vn2-vi2)  -2(vnwncos  fi-viwi  cos  a).    (9) 
Now  from  eqs.  (5)  and  (6)  we  easily  find,  by  subtraction, 

Pi     pn    wn2-w^ 

J-J       -%j-    +hi+hn,       ....     (9) 

in  which  hi+hn  may  be  replaced  by  h.  Between  eqs.  (9)  and 
(9')  wn2—  wi2  is  eliminated,  whence,  noting  that  w\  .  cos  a  =ui 
and  wn  .  cos  p=un  (Fig.  23),  we  have 


But  from  eq.  (10)  of  §  34  the  work  done  (per  second)  by  the 
couple  to  which  the  water's  action  on  wheel  is  equivalent  is 


§  42.          BERNOULLI'S  THEOREM.     ROTATING  CASING.  59 

Qr 


,  ......     (11) 


which  in  this  case  (without  friction)  must  =  #V  (see  Fig.  24). 
We  also  have  Qj-h  =  Rfvf  (see  eq.  (4)  of  §  38);  whence  it  follows 
that 


This  being  substituted  in  eq.  (10)  there  results 
^P-_£+2l  +  ^=*!)f 

2g     r    2g     r        -g 

which  is  known  as  Bernoulli's  theorem  (without  friction;  for 
steady  flow  of  water  in  a  (uniformly)  rotating  casing  (rotating 
around  a  vertical  axis).  It  is  noticeable  that  it  does  not  con- 
tain the  absolute  velocities  of  the  water  at  entrance  and  exit  of 
a  channel,  but  only  the  relative  velocities  of  the  water,  the  fluid 
pressures,  and  the  velocities  vi  and  vn  of  the  two  wheel-rims 

(v  2_V2\ 
-^—^  -  )  is  sometimes  called  the  cen- 

trifugal head. 

42.  Bernoulli's  Theorem  (Rotating  Casing)  when  Friction  "is 
Considered.  —  If  eqs.  (5a)  and  (6a)  be  combined  we  find 


(9,a) 


in  which  hi  +  hn  may  be  replaced  by  h. 

This  may  now  be  combined  with  (9)  to  eliminate  (wn2—Wi2), 
remembering  that  wi  cos  a  =  u\  and  wn  cos  /j.  =  un,  whence  we 
have 


n       j 

g       r    r       2g  g 

Now  in  the  derivation  of  eq.  (10),  §  34,  for  the  power  due  to 
the  action  of  the  "  equivalent  couple7'  of  water  on  wheel,  the 
forces  dealt  with,  of  water  prisms  on  the  wheel-blades,  were  the 
actual  forces,  including  frictional  components,  if  any.  Hence 
that  equation  will  still  stand  as  to  its  form,  now  that  we  are  con- 
sidering friction.  The  equation  is 


60  HYDRAULIC    MOTORS.  §  42a. 


We  also  have  eq.  (4a)  of  §  40,  viz., 

Qr(h  -hf-  h"  -  h'")  =  R'v'  +  R"v", 

for  the  case  where  friction  is  considered,  and  note  that  the  power 
given  by  eq.  (lla)  above  is  expended  on  Rf  and  R",  t,hat  is, 

R'V'  +  R"V"  =  %%Vl  -  UnVn)  ; 

y 
hence 


unvn]  +  g)',   .     .     .     (12a) 
and  this  value  of  h,  placed  in  eq.  (10a)  above,  gives 

^+P2=fj!+Pl+^L2_^    .    .    .     a3fl) 
2#       T     2g       r  2g 

which  is  Bernoulli's  theorem  for  steady  flow  in  a  rotating  casing 
when  friction  is  considered.  It  is  seen  that  the  quantity  h"  is 
what  was  called  the  "loss  of  head  occurring  in  the  wheel-chan- 
nel/' so  that  (13a)  differs  from  (13)  only  in  the  introduction  of 
this  loss  of  head. 

Here  we  note  again  that  the  absolute  velocities  of  the  water, 
at  points  1  and  n,  entrance  and  exit  of  a  wheel-channel,  do  not 
appear  in  this  theorem,  but  simply  the  relative  velocities,  the 
"pressure-heads,"  the  "  centrifugal  head/'  and  the  loss  of  head 
h". 

42a.  Turbine  Pump.  —  If  it  is  required  in  Fig.  24  that  the 
direction  of  the  flow  of  water  be  reversed  ;  that  is,  that  a  steady 
flow  of  water  is  to  be  maintained  from  the  lower  level  B  to  the 
upper  level  A,  with  steady  operation  of  a  properly  designed 
rotating  "pump"  (as  it  now  becomes),  or  reversed  turbine, 
having  properly  curved  channels,  but  with  inlet  1  communicating 
with  B  and  outlet  n  with  A*  ;  it  is  evident  that  all  the  relations 
of  the  previous  paragraph  still  hold  good  with  these  differences: 
Instead  of  a  resisting  force  R'  we  must  have  a  working  force  P, 
and  the  cable  must  unwind  from  the  drum  instead  of  being 
wound  up.  The  working  force  will  have  to  be  furnished  by  some 

external  source  of  power,  and  if  the  velocity  of  the  cable  be  now 

*  See  next  page. 


§  42a.        BERNOULLI'S  THEOREM.     ROTATING  CASING.  61 

called  v,  we  have  for  the  case  where  no  loss  of  head  occurs  in 
any  part  between  B  and  A  (and  no  axle  friction  of  pump) 


whereas,  if  losses  of  head,  etc.,  occur  (using  same  notation  as  in 
§§  38  to  42), 

Pv  =  Qf[h  +  hf +h" +h'"]  +  R"vff,   .     .     .     .     (15) 
instead  of  eq.  (4a) . 

Also,  Bernoulli's  Theorem  for  steady  flow  in  a  rotating  pipe 
revolving  uniformly  in  a  horizontal  plane  remains  the  same  as. 
(13a),  viz., 

2  2       /n  2  2 

^•+7=2^+7+~ ^*r-h">  •  •  •  (16) 

provided  the  flow  (relative)  is  still  from  1  to  n. 


*  Or,  Fig.  24  may  be  conceived  to  be  changed  in  this  respect :  that  B  is 
still  the  receiving-tank,  and  A  the  source  of  supply,  but  that  B  is  at  a  higher 
elevation  than  A. 


CHAPTER  IV. 

IMPULSE  WHEELS. 

43.  Definition  of  Impulse  Wheels. — Water-wheels  furnished 
around  the  rim  with  small  buckets,  or  cups,  or  curved  vanes 
closed  in  on  the  sides,  and  receiving  the  action  of  a  "'free  jet" 
of  water  directed  tangentially  to  the  rim  or  nearly  so,  are  called 
"Impulse  Wheels7';    sometimes  "tangential  wheels".     By  a 
"free  jet"  is  meant  one  which  is  formed  "free"  in  the  atmos- 
phere, flowing  from  the  extremity  of  a  nozzle,  often  of  a  con- 
verging conical  shape,  though  sometimes  of  rectangular  cross- 
section. 

We  shall  first  consider  that  the  greatest  radial  width  of  each 
cup  or  bucket  is  small  compared  with  the  radius  of  the  rim  of 
the  wheel,  in  which  case  the  movement  of  a  cup  may  be  con- 
sidered to  be  one  of  translation. 

44.  Pressure  of  Free  Jet  upon  a  Fixed  Solid  of  Revolution, 
when  the  Axis  of  the   Solid  is  Coincident  with   the  Axis  of  the 
Jet. — See  Fig.  26.     Here  the  jet  is  deviated  smoothly  and  sym- 

C 


FIG.  26. 


metrically  on  all  sides  of  the  axis,  OX,  of  the  jet;   OX  is  also 
the  axis  of  the  -fixed  solid  AB.     The  filaments  of  the  jet  meet 

62 


§  44.  IMPULSE  WHEELS.  63 

the  surface  of  the  solid  tangentially,  the  center  of  the  latter 
being  pointed,  as  shown.  The  particles  of  the  water,  if  we 
neglect  friction,  have  the  same  velocity  on  leaving  the  outer 
edge  of  the  solid  at  A,  or  B,  as  they  had  on  leaving  the  tip  of 
the  nozzle  at  F,  viz.,  c  ft.  per  second;  but  the  directions  of 
their  motion  at  A,  being  tangent  to  the  solid,  make  an  angle 
a  with  the  original  direction  OX.  This  angle  is  evidently  the 
same  in  value  for  all  particles  as  they  pass  off  the  solid  at  A, 
the  edge  of  the  circle  whose  diameter  is  AB  and  whose  plane 
is  perpendicular  to  OX. 

The  resultant  pressure,  P  Ibs.,  of  the  jet  against  the  solid 
during  this  steady  flow,  is  found  by  considering  that  in  a  small 
time  At  seconds  a  small  mass  Am  of  water  has  had  its  velocity 
in  the  direction  of  OX  diminished  from  a  value  of  c,  to  c  cos  a,  ft. 
per  sec.  Hence  a  force  equal  and  opposite  to  the  force  P 

c— ccos  a  . 
has   occasioned  a  negative  acceleration  p= -r in  the 

component  of  velocity  parallel  to  OX,  of  the  mass  Am. 

Vc  —  c.  cos  a~\ 

.'.  P,=massXaccel.,=Am\ -j- I.      .     .     (1) 

Now  if  Q  is  the  volume,  per  second,  of  water  issuing  from  the 
nozzle  (being  also  the  volume  per  second  acting  on  the  solid; 
since  the  latter  is  held  at  rest),  we  have  for  the  mass  passing 


over  it  hi  At  seconds  AM  =  [  — 


(~\At,  and  therefore  may  write 

\   u    / 

Ore 
P=—  (1-cosa)  .......     (2) 

y 

For  exampk,  with  a  jet  of  one  inch  diameter  having  a  velocity 
of  40  ft.  per  sec.,  the  angle  a  being  45°,  we  have 


=j  X  40  =  0.218  cub.  ft.  per  second; 


64  HYDRAULIC   MOTORS.  §  45, 

45.  Pressure  of  Free  Jet  on  Solid  of  Revolution  when  the 
Latter  is  in  Motion  away  from  the  Jet. — As  before,  the  solid  is 
one  of  revolution  with  its  axis  coinciding  with  that  of  the  jet 
and  its  motion  is  assumed  to  have  a  uniform  velocity  v  (less 
than  that,  c,  of  the  water  in  the  jet)  and  to  be  directed  along 
the  axis  OX.  (See  Fig.  27.) 


Here,  for  a  given  volume  per  second  Qf  passing  over  the 
solid  the  resultant  pressure  P  of  the  water  against  the  solid 
will  be,  of  course,  less  than  before,  for  the  same  angle  a;  but 
since  the  solid  is  now  in  motion  P  is  a  working  force,  for  it; 
and  to  prevent  acceleration  of  the  motion  of  the  solid  a  resist- 
ance R  Ibs.  equal  to  P  and  in  same  line,  but  oppositely  directed, 
is  supposed  to  be  furnished.  It  is  required  to  find  the  value 
of  P  for  a  given  v  and  c  and  sectional  area,  F  sq.  ft.,  of  the  jet. 

The  velocity  of  a  particle  of  water  is  c  just  before  encounter- 
ing the  point  of  the  solid.  On  leaving  the  further  edge  of  the 
solid,  as  at  A,  its  velocity  relatively  to  the  solid  (since  friction 
is  neglected  and  A  is  not  appreciably  higher  or  lower  than  the 
nozzle)  is  the  same  as  before,  viz.,  c— v;  but  the  direction  of 
this  relative  velocity  is  at  an  angle  a  with  the  former  direction 
0 .  .X,  and  the  point  A  has  itself  a  velocity  v  parallel  to  0 .  .X. 
Hence  the  absolute  velocity  (i.e.,  relatively  to  the  earth)  is 
w  =  (Aw  in  figure),  the  diagonal  of  the  parallelogram  formed 
on  the  relative  velocity,  c— v/ and  AH,=v,  as  sides.  Hence 
the  loss  of  velocity  of  the  particle  in  the  direction  0 ...  X  is 
equal  to  c  diminished  by  AC,  the  projection  of  w  on  OX.  But 
evidently  this  projection,  or  velocity-component,  AC  is  made 
up  of  v  and  HC,  PIC  being  equal  to  (c  —  v)  cos  a. 


§  46.  IMPULSE   WHEELS.  65 

Therefore  the  loss  of  velocity,  in  direction  0,.  .X,  of  the 
small  mass  AM  passing  over  the  solid  in  the  time  At  is 

c  —  [v  +  (c  —  v)  cos  a],     or     (c  —  v)(l—  cos  a). 

(c  —  v)(l—  cos  a) 
Ac  before,    P  =  massXaccel.  =  AM.  .  -          ^         —  . 

Q'r 
But  the  value  of  AM  is  —  ^At,  where  Q'  is  the  volume  of 

*J 

water  passing  per  second  over  the  solid  (and  not  that,  Q,  issuing 
from  the  nozzle)  .     Hence 


and  the  work  done  by  this  working  force  on  the  solid  every 
second  is 


o 

ft.-lbs.  per  sec.,  and  is  expended  in  overcoming  the  resistance 
R  through  v  ft.  each  second;  that  is,  Pv=Rv.  Evidently  if  a 
is  made  greater  than  90°,sthe  solid  of  revolution  becomes  a 
cup,  concave  to  the  jet. 

46.  Impulse  Wheels.  —  It  is  to  be  specially  noted  that  in 
eq.  (4)  Qf  denotes  the  volume  (say  cub.  ft.)  passing  per  second 
over  the  solid  of  revolution,  so  that  Q'=F(c—v);  and  not 
Fc,  =  Qj  which  is  the  volume  per  sec.  issuing  from  the  nozzle. 
But  if  a  motor  be  constructed  consisting  of  a  series  of  such 
solids  of  revolution,  or  cups,  or  of  equivalent  curved  vanes, 
coming  into  position  successively  and  endlessly,  which  would 
be  the  case  if  they  were  placed  on  the  rim  of  a  wheel  of  large 
radius,  more  work  per  second  could  be  done  and  in  proportion 
to  the  water  used;  since  more  than  one  solid  or  cup  would  be 
in  action  at  certain  times,  the  portion  of  jet  intercepted  be- 
tween two  consecutive  cups  being  able  to  finish  its  action  on 
the  cup  in  front,  while  new  work  is  being  done  on  the  adjacent 
hinder  cup.  With  this  arrangement,  therefore,  all  the  water 
issuing  from  the  nozzle  would  be  used,  and  for  Q'  we  may  sub- 
stitute Q  and  thus  obtain  for  the  power  of  a  motor  provided 
with  such  a  series  of  cups  the  expression 


fctt  HYDRAULIC    MOTORS.  §  40, 

T     Qr(c—v)[l  —cos  a]v 

9 

ft.-lbs.  per  sec.     The  corresponding  average  working  force  is 
p_Qr(c-v)(l-cosa) 

9 

Ibs.  It  is  evident  from  (5)  that  for  a  given  water-supply,  Q  cub. 
ft.  per  second,  the  value  of  the  power  L  depends  on  both  v  and 
the  angle  a,  becoming  zero  both  for  v  =  zero  (stationary  cup) 
and  for  v  =  c  (in  which  case  the  jet  does  not  overtake  the  cup). 
It  is  also  zero  for  a  =  0°. 

For  any  constant  v,  L  is  evidently  a  maximum  for  a  =  180°, 
i.e.,  for  cos  a  =  —  1 ;  and  then  takes  the  form 

2Qr(c-v)v  ,,  . 

L==  — - (5a) 

This  value  of  the  angle  a  may  be  attained  by  giving  to 
the  solid  of  revolution  the  form  of  a  ring-shaped  cavity  the 
tangents  to  whose  outer  rims  are  parallel  to  the  jet  so  that 
both  the  relative  velocity  c— v,  and  absolute  velocity  w,  of 
the  water  leaving  the  solid  (or  cup,  as  it  may  now  be  called) 
are  parallel  to  the  original  jet.  The  pointed  center  of  the 
cup  provides  for  a  gradual  deviation  of  the  water  from  its  original 
path  and  prevents  eddying  and  consequent  internal  fluid 
friction.  (See  Fig.  28.)  When  a  seri.?s  of  such  cups  is  fastened 

on  the  edge  of  a  large  wheel, 
however,  the  center  of  each 
cup  does  not  remain  accurately 
in  the  axis  of  the  jet  when 

1  'I'E^  under  its  action,  since  this  center 

>.)w%y  is  moving  in  the  arc  of  a  circle. 

For  the  point,  therefore,  a  sharp 
ridge  is  substituted  whose  edge 

lies  in  the  plane  of  rotation,  thus  providing  for  a  gradual  devia- 
tion of  the  water  at  all  times  during  the  action  of  the  jet  on 
any  given  cup.  This  dividing  ridge  separates  the  cup  or  bucket 
into  two  lobes,  thus  giving  rise  to  the  general  form  adopted 


FIG.  30.    5,000  H.P.  PKI/TON  WHEEI,. 

Thejabove  wheel,  9  feet  iq  inches  in  diameter,  is  capable  of  developing  5,000  H.P.  at  225 
R.P.M.,  when  operating  under  r6s  feet  effective  head. 


§47. 


IMPULSE    WHEELS. 


67 


V 


in  the  Pelton  and  Dob.e  impulse  wheels,  as  shown  in  Figs.  30 
and  31  (opposite  pp.  67  and  69). 

Fig.  29  represents  a  simple  impulse  wheel  of  this  kind.  A 
resistance,  Rf  Ibs.,  is  shown,  acting  tangent  to  the  edge  of  a 
pulley  of  radius  r'  keyed  upon  the  shaft  of  the  water-wheel. 
Without  such  resistance,  of 
course,  the  wheel  would  "  speed  J"t 
up"  until  the  velocity  of  the  ^^ 
cups  reached  a  value  equal  to  zl£i 
that,  c,  of  the  water  in  the  jet.  ^^ 
The  working  force  would  then  u 
disappear,  and  while  a  display  |~" 
of  high  speed  might  be  made, 
no  power  would  be  obtained, 
the  jet  passing  on  as  if  the  wheel 
were  not  present. 

It  still  remains  to  determine 
the  special  value  of  v,  the  cup 
velocity,  for  which  the  power,  L, 
is  a  maximum.  Since  from  eq.  (5a) 


I/=(a  constant )X^c-v) v, 
we   find  that  by  putting  -77  equal  to  zero,  i.e.,  c  —  2v  =  Q,  a 

/•» 

value  of  v  =  —  gives  a  maximum  L.     The  substitution  of  this 

special  value  of  v  for  the  v  of  eq.  (5a)  results  in  the  following 
expression  for  the  maximum  power,  viz., 


a 


c  \c     Qf    c2 
•r~J-2 


(7) 


47.  Efficiency  of  the  Impulse  Wheel. — It  is  to  be  noted  that 
in  eq.  (7)  Qr  +  gi$  the  mass  of  water  flowing  per  second  from  tho 
nozzle  and  c  =  \/2gh  if  there  is  no  friction  at  edges  of  the  nozzlf 
(and  hi  be  considered  equal  to  h,  Fig.  29),  so  that  L^Qfh,  theoret- 
ically; from  which  it  is  seen  that  the  efficiency  of  this  wheel,  run 
at  proper  speed,  should  be  100  per  cent,  if  it  were  of  perfect 


68  HYDRAULIC    MOTORS.  §'47. 

construction  and  if  all  friction  could  be  avoided.  This  is  as  it 
should  be,  since  the  absolute  velocity  of  the  water  as  it  leaves  the 
outer  rim  of  a  bucket  [this  velocity  having  in  general  a  value  = 
Velocity  of  cup  —relative  velocity  of  water  at  rim,  i.e.;  =  v  —  (c  —  v\ 

g 
or  2v  —  c]  would  in  this  case  be  2X-^-c;    =zero.      In  other 

words,  the  water  possesses  no  kinetic  energy  on  leaving  the 
cup.  At  the  beginning  of  its  path  on  the  cup  it  -has  kinetic 
energy,  but  no  potential  nor  pressure  energy  (i.e.,  none  above 
that  due  to  atmospheric  pressure),  and  at  exit  none  of  any 
kind.  •  It  has  given  up  its  whole  stock  of  energy. 

On  account  of  imperfect  guidance  of  the  water  by  the  walls 
of  the  bucket  and  the  friction  of  the  water  on  itself  and  on  the 
surfaces  of  the  bucket,  (aside  from  the  fact  that  the  value  of  the 
angle  a  cannot  be  made  exactly  180°,)  the  efficiency  of  theso 
wheels  is  brought  down  in  practice  to  values  ranging  from  70 
to  90  per  cent.,  according  to  circumstances.  (See  test  quoted 
on  pp.  809  and  810,  M.  of  E.) 

In  Fig.  29  the  head  hi  is  the  depth  of  still  water  just  behind 
the  center  of  the  nozzle;  but  in  practice  a  conical  nozzle  is- 
generally  employed,  attached  to  a  pipe  or  other  source  of  steady 
supply,  and  the  efficiency  of  the  wheel  is  generally  referred  to 
the  total  head  (above  atmosphere)  at  the  base  of  the  nozzle 
where  the  pressure  is  (say)  p\  Ibs.  per  square  inch  (above  tho 
atmosphere)  and  the  velocity  (much  less  than  that  .of  the  jet;. 
since  the  sectional  area  F\  is  much  larger  than  that  of  the  jet) 
is  ci. 

In  such  a  case  we  may  write,  therefore,  in  place  of  hi 


+        and  hence 


if  (/>,  =0.95,  is  the  coefficient  of  the  nozzle.  If  the  efficiency 
is  80  per  cent,  (for  instance),  we  have  for  the  useful  power 
(neglecting  axle  friction) 


FIG.  31.     Doble  Impulse  Wheel. 


35.     Jet  from  Doble  Nozzle. 


§  48.  IMPULSE  WHEELS.  69 

48.  Numerical   Example,   Impulse    Wheel. — Compute  what 
resistance,  R' ' ,  can  be  continuously  overcome,  tangent  to  a  pul- 
ley of  a  radius  r'  =  2  ft.  keyed  upon  the  shaft  of  a  Pelton  wheel, 
whose  cup  centers  form  a  circle  of  r  =  3  ft.  radius;  if  the  avail- 
able water-supply  is  Q  =  1.50  cub.  ft.  per  sec.,  issuing  in  a  "free 
jet "  from  a  conical  nozzle  at  whose  base  the  fluid  pressure  is 
measured  and  found  to  be  pi  =  130.2  Ibs.  per  sq.  in.  (above  the 
atmosphere).     The  diameter  of  the  base  of  the  nozzle  is  di=4. 
inches,  and  that  of  the  part  of  jet  where  its  filaments  have  become 
parallel  is  d=1.44  in.     The  wheel  is  to  be  run  at  best  speed 
und  an  efficiency  of  80  per  cent,  is  counted  on. 

70  -^  o 

Solution. — ci  =  Q-±—7~  =  17.2  ft.  per  sec.,  and  hence  —  =  4.6 

ft.  Also  c  =  Q  +  -  =  133  ft .  per  second ;  while  — ,  =  — ~  9  r — , 
-300  ft.  We  may  therefore  compute  the  coeff.  0  from  c  = 
obtaining  0  =  0.95,  the  "coefficient  of  the 

nozzle."     Substitution  in  eq.  (8)  now  results  as  follows: 
72V  =  0.80 X  1.5 X62.5[300  +  4.6]  =  22,845  ft. -Ibs.  per  sec.; 

or  41.5  horse-power. 

The  proper  speed  of  the  cups  or  buckets  should  be  v  =  c  +  2, 

=  66.5  ft.  per  sec.     The  value  of  v'  is  •§•  of  v  and  =  44.3  ft.  per 

sec.     Finally,  therefore,  for  R'  we  have 

R'  =  L  +  v'  =  22845  -j-  44.3  =  51.6  Ibs. 

This  force  Rf  may  be  the  tension  in  a  cable  winding  upon  a 
drum  or  the  tangential  component  of  the  pressure  between  the 
teeth  of  a  pinion  on  another  shaft  and  those  of  a  gear-wheel 
on  the  shaft  of  the  Pelton  wheel.  Of  course,  if  the  radius  / 
of  the  drum  or  gear-wheel  is  changed,  the  value  of  R'  will  be 
altered  i~i  inverse  ratio. 

49.  Flat  Plates  instead  of  Cups. — If  flat  plates  were  substituted 
for  the  cups  of  the  impulse  wheel,  the  highest  theoretical  power 
would  be,  as  with  the  ordinary  undershot,  only 

Or 
L  =  ^-(c-v)v; (9) 

<J 


70  HYDRAULIC    MOTORS.  §  50. 

which,  with  v  =  c+2  for  best  effect,  would  give  for  the  power 


....     (10) 

as  the  highest  theoretical  performance;  reduced  in  practice  to 
25  to  35  per  cent,  efficiency,  in  place  of  the  0.50  of  eq.  (10). 

50.  American  Impulse  Wheels.  —  Early  in  the  second  hal"  of 
the  nineteenth  century  simple  impulse  wheels  were  constructed 
in  California  provided  with  flat  plates  as  buckets.  These  so- 
called  "  Hurdy-gurdies/'  though  of  low  efficiency,  were  easily 
and  cheaply  made,  and  the  speed  of  rotation  could  be  easily 
varied  by  a  change  of  radius.  The  substitution  of  approxi- 
mately hemispherical  cups  for  the  flat  plates  brought  about  a 
great  improvement  in  performance,  and  later  the  invention 
of  the  dividing  ridge,  the  main  feature  of  the  Pelton  bucket, 
raised  the  efficiency  to  a  high  figure;  and  this  improved  type 
of  impulse  wheel  is  now  widely  used  throughout  America  and 
Europe. 

The  three  principal  forms  of  impulse  wheel  with  buckets 
characterized  by  the  dividing  ridge,  or  its  equivalent,  as  made 
in  the  United  States,  are  those  manufactured  by  the  Pelton 
Water-wheel  Co.  of  San  Francisco  and  New  York,  the  Abner 
Doble  Co.  of  San  Francisco,  and  the  James  Leffel  Co.  of  Spring- 
field, Ohio.  Perspective  views  of  the  three  wheels  made  by 
these  companies  are  shown  in  Figs.  30,  31,  and  32,  opposite  pp. 
67,  69,  and  70,  respectively,  of  this  book.  As  will  be  seen 
from  these  representations,  the  two  lobes  of  the  Pelton  bucket 
are  rectangular  in  form,  while  those  of  the  Doble  wheel,  called 
"  ellipsoidal  "  by  the  makers,  are  oval,  with  notches  cut  out 
at  the  point  of  first  impingement  of  the  jet.  In  the  "  Cascade" 
wheel  made  by  the  James  Leffel  Co.,  the  "lobes,"  or  half- 
buckets,  are  set  "staggering,"  or  "breaking  joint,"  on  the 
two  sides,  and  near  the  rim,  of  a  thin  circular  disc,  whose  sharp 
edge  serves  the  same  purpose  as  a  dividing  ridge  to  split  the 
jet.  Fig.  33  (opposite  p.  72)  shows  the  Escher-Wyss  type  of 
impulse  wheel,  made  in  America  by  the  Allis-Chalmers  Co. 
of  Milwaukee. 


FIG  32.     The  "Cascade"  (Leffel)  Impulse  Wheel. 


§  51.  IMPULSE   WHEELS.  7l' 

51.  Regulation  of  Pelton  Impulse  Wheels.— A  conical  nozzle1 
(one  or  more)  furnishing  a  cylindrical  jet  of  circular  cross-section 
is  generally  employed  with  impulse  wheels  of  the  Pelton  type. 
At  the  base  of  the  cone  the  pressure,  pif  is  high  and  the  velocity r 
d,  is  small,  in  regular  running;  the  energy  being  chiefly  in* 
the  pressure  form  at  that  point  of  the  flow.  When  the  re-- 
sistance  R'  is  reduced  below  its  usual  value,  unless  the  work- 
ing force  P  on  the  buckets  is  reduced  in  the  same  proportion, 
the  velocity  of  the  wheel  will  be  accelerated;  and  this  is  usually 
undesirable,  especially  in  the  running  of  electric  generators. 
A  reduction  of  P  can  be  effected  by  reducing  the  size  of  the 
jet,  or  by  reducing  its  velocity,  or  by  diverting  the  direction 
of  jet  sufficiently  so  that  only  a  portion  acts  on  the  buckets. 

To  reduce  the  velocity  requires  a  reduction  in  the  value 
of  pi  at  the  base  of  the  nozzle;  and  this  is  frequently  effected 
by  the  partial  closing  of  a  valve-gate  in  the  pipe  just  up-stream 
from  the  nozzle.  But  this  necessitates  a  loss  of  head  in  the 
supply-pipe  due  to  the  sudden  enlargement  of  section  ex- 
perienced by  the  water  in  passing  froni  the  narrow  section- 
under  the  valve-gate  to  the  full  section  of  the  pipe,  and  the 
efficiency  of  the  wheel  is  much  reduced.  This  loss  of  efficiency 
is  due  to  two  causes:  First,  the  jet  velocity  and  that  of  the 
bucket  no  longer  have  the  proper  relation  for  best  effect. 

f)\       Ci^ 

Secondly,  the  effective  head,       +  %->  (piis  here  the  pressure  in 

excess  of  the  atmosphere,)  at  the  base  of  the  nozzle,  has  less 
than  its  usual  value.  This  so-called  "throttling"  of  the 
flow  to  produce  the  diminution  of  jet-velocity  is  therefore  a 
very  wasteful  expedient  in  cases  where  economy  in  the  use 
of  water  is  of  importance. 

Diversion  of  the  jet  (so  that  only  a  portion  acts  on  the 
buckets)  without  throttling  is  also  wasteful  of  water,  though 
often  resorted  to  in  situations  where,  on  account  of  the  ex- 
treme length  of  the  supply-pipe,  a  checking  of  the  flow  would 
produce  dangerous  "water-hammer  "  (see  later,  in  §  125). 

To  diminish  the  value  of  the  working  force  P  without 
materially  altering  the  jet-velocity,  and  thus  retain  the  proper 


72  HYDRAULIC    MOTORS.  §  52. 

relation  between  the  latter  and  that  of  the  buckets,  requires 
a  reduction  in  the  sectional  area  of  the  jet.  A  common  way 
of  doing  this  at  the  present  day,  with  impulse  wheels  of  the 
Pel  ton  type,  is  by  the  use  of  an  internal  conical  "  stopper," 
or  "spear-head/'  of  brass,  frequently  called  a  "needle,"  par- 
tially closing  the  base  of  the  conical  nozzle  and  capable  of 
longitudinal  movement.  Fig.  34  shows  a  longitudinal  section 


POILE    NEEDLE    ROUTING    HOZZLC. 
FIG.    34. 


of  the  Doble  Needle  Regulating-nozzle  as  used  with  the  Doble 
impulse  wheel.  The  water  passes  through  the  space  AB 
(and  CD)  toward  the  left.  By  the  advance  of  the  "needle" 
E  toward  the  left  the  ring-shaped  space  between  it  and  the 
edge  of  the  nozzle  opening  is  progressively  diminished  in  sec- 
tional area. 

The  filaments  of  water  converge  toward  and  beyond  the 
point  of  the  "needle  "  and  finally  forrn  a  solid  cylindrical  jet 
of  circular  section,  "a  clear,  transparent,  polished  stream," 
in  the  words  of  a  thesis  by  Messrs.  H.  C.  Crowell  and  G.  C.  D. 
Lenth  of  the  Mass.  Inst.  of  Technology,  published  in  June 
1903.  Fig.  35  (opposite  p.  69)  is  from  a  photograph  of  a  jet 
issuing  from  one  of  these  Doble  nozzles,  and  is  taken  from  the 
thesis  mentioned.  The  size  of  the  jet  depends  on  the  position 
of  the  "needle,"  and  not  on  the  head  of  water 

52.  Girard  Impulse  Wheels,  or  "  Impulse  Turbines." — An- 
other form  of  impulse  wheel  may  be  formed  by  two  flat,  parallel, 


FIG.  33.     Escher,  Wyss  &  Co.     Impulse  Wheels. 


IMPULSE   WHEELS. 


73 


and    concentric    rings, 


crowns/'7   between   which   are   in- 


serted numerous  curved  blades,  or  "vanes"  (sometimes  bent 
from  flat  plates  and  therefore  cylindrical);  and  is  shown  in 
vertical  and  horizontal  sections  in  Fig.  36.  The  wheel  receives 
water  in  a  "free  jet"  from  a  nozzle  fitted  to  a  pipe  P,  placed 
either  on  the  inside  of  the  wheel,  as  in  this  figure  (and  then 
the  wheel  is  called  an  "  out  ward-flow  impulse  wheel),  or  on 
the  outside  (an  "inward-flow"  wheel). 

This  form  of  wheel,  now  called  in  Europe  a  "  Girard  Impulse 
Wheel/'  was  invented  by  Poncelet  in  1826  and  first  applied 
practically  by  Zupinger.  The 
wheel  revolves  in  the  direction 
shown  in  the  figure,  and  the 
water  in  passing  through  it 
along  a  vane  does  not  fill  the 
entire  space  between  the  two 
consecutive  vanes  and  is 
therefore1  exposed  to  atmos- 
pheric pressure  on  one  side 
throughout  its  whole  course. 
The  shapes  and  positions  o£ 
vanes  are  to  be  so  designed, 
and  a  proper  speed  of  wheel 
so  determined,  as  to  give  a 
power*  (72V)  to  be  expended 
in  overcoming  some  constant 
resistance,  R' ' ,  tangent  to 
some  pulley  or  gear-wheel 
(keyed  upon  the  same  shaft  as  the  water-wheel),  the  velocity 
of  a  point  in  whose  outer  edge  is  vf  ft.  per  sec. 

The  deviation  of  the  water  of  the  jet  from  its  original  direc- 
tion, and  the  progressive  reduction  of  its  absolute  velocity, 
are  accomplished  gradually  after  entrance  upon  the  vane; 
and  each  particle  is  considered  to  move  in  a  plane  parallel 
to  the  crown  plates,  the  vertical  thickness  of  the  jet  being 
equal  to  the  distance  apart  of  those  plates  (m  and  n  in  Fig 
36)  =e. 

*  Maximum  power. 


FIG.  36. 


N 


'•n 


FIG.   38- 


74 


§  53.  GIRARD    IMPULSE   WHEELS.  75 

53.  Best  Speed  of  the  Girard  Wheel. — In  the  horizontal 
section  of  a  portion  of  an  outward-flow  Girard  wheel  shown  in 
Fig.  37,  the  water  of  the  jet  entering  at  point  1  has  an  abso- 
lute velocity  of  w\  ft.  per  sec.,  at  an  angle  of  a  with  a  tangent, 
1 .  .  Tj  to  the  inner  wheel-rim,  the  velocity  of  this  inner  rim  itself 
being  constant  at  some  value  v\.  Since  the  jet  is  a  free  jet, 
the  value  of  Wi  is  not  dependent  (beyond  a  slight  extent)  on 
the  presence,  or  velocity,  of  the  wheel. 

With  Wi  as  the  diagonal  of  a  parallelogram  of  velocities 
(p.  87,  M.  of  E.),  and  Vi  as  one  side  (both  springing  from  the 
point  1),  a  parallelogram  is  constructed  whose  other  side,  c\, 
will  be  the  relative  velocity  of  the  water  at  1,  i.e.,  relatively 
to  that  point  of  the  inner  rim  of  the  wheel) .  We  assume  that 
whatever  the  "'best  speed"  of  the  wheel  proves  to  be,  the  tan- 
gent at  1  to  the  wheel- vane  1.  .N  is  made  to  coincide  with  the 
direction  of  c\,  the  relative  velocity,  making  some  angle  /? 
with  v\j  i.e.,  with  the  rim-tangent  1.  .T. 

In  this  way  the  water  will  glide  smoothly  upon  the  vane 
1.  .N  without  "shock  "  or  eddying  (which  is  always  to  be  avoided; 
since  it  causes  waste  of  energy).  The  vane  is  curved  back- 
ward from  1  to  N  so  as  to  produce  (if  its  motion  is  not  too  rapid) 
a  deviation  of  the  motion  of  the.  water  particles  from  the  rec- 
tilinear path  they  would  otherwise  pursue  into  a  curved  path, 
1.  .N'  (absolute  path).  This  deviation  is  accompanied  by 
a  gradual  diminution  of  the  absolute  velocity  of  the  water. 
(If  the  vane  were  stationary,  there  would  be  practically  no 
change  in  the  absolute  velocity.)  On  the  arrival  of  the  water 
particles  at  the  outer  rim  of  the  wheel  the  outer  extremity  N 
of  the  vane  has  come  to  the  position  N'.  The  relative  velocity 
at  N'  (i.e.,  relatively  to  the  outer  end  of  vane)  has  now  a  differ- 
ent value,  cnj  from  that  at  the  point  of  entrance  and  is,  of 
course,  tangent  at  N'  to  the  vane  curve.  The  point  AT'  of  the 

M 

outer  rim  of  wheel  has  a  velocity  vn  equal  to  —v\  (from  the 

proportion  Vi:vn:  :ri:rn);    and  a  parallelogram  formed  on  vn 
and  cn  as  two  sides,  will  determine  the  value  of  wn,  the  abso- 


76  HYDRAULIC   MOTORS.  §  54 

lute  velocity  of  the  water  at  exit,  as  being  the  diagonal  N'C' 
of  this  parallelogram. 

It  is  seen  that  wn  is  much  smaller  than  the  corresponding 
value  wi  at  entrance.  Denote  by  d  the  angle  between  cn  and 
a  line,  N'T',  drawn  tangent,  at  N',  to  the  outer  rim  of  wheel. 

To  determine  the  best  value  (i.e.,  conducive  to  greatest 
efficiency)  for  the  velocity  v\  (or  vn)  we  must  note  that  the 
kinetic  energy  carried  away  per  second  by  the  water  at  exit, 

Or  w  2 
viz.,  -  —  '—jjj-j  should  be  as  small  as  possible;  and  this  means 

that  wn  should  be  as  small  as  possible.  Inspection  of  the 
parallelogram  of  velocities  at  Nf  shows  that  a  small  value  for 
the  angle  d,  and  equality  between  vn  and  cn,  conduce  to  a 
small  wn.  Now  the  angle  d  cannot  be  made  equal  to  zero, 
but  may  usually  be  made  as  small  as  15°;  it  is  quite  feasible, 

however,  to  have  cn  =  vn  ;     ........     (1) 

which  assumption  will  therefore  now  be  made  and  the  result 
noted. 

Bernoulli's  Theorem  without  friction  (eq.  (13),  §  42)  may 
now  be  applied  to  the  steady  flow  between  1  and  N  in  the 
rotating  channel  here  presented  (it  being  noted  that  the  water 
is  under  atmospheric  pressure  both  at  1  and  N  so  that  the 
pressure-heads  cancel  out),  leaving 

c^-ci'-uw1-*!1;    ......    (2) 

in  which  if  we  put  cn=vnirom  eq.  (1)  there  results  Ci=vi, 
which  shows  that  the  parallelogram  at  point  1  must  be  made 
a  rhombus.  Hence,  from  trigonometry, 


as  the  best  value  for  the  inner  wheel-rim  velocity.  The  re- 
sistance R'  should  therefore  have  a  corresponding  value  (in  con- 
nection with  v')  to  prevent  acceleration  of  the  velocity  beyond 
this  speed.  Evidently  /?  must  be  made  equal  to  2a. 

54.  Power  of  the  Girard  Impulse  Wheel.  —  The  power  of  the 
wheel  due  to  the  action  of  the  water,  at  this  special  speed,  is 


§  54.  GIRARD    IMPULSE    WHEELS.  77 

now  obtained  by  using  the  formula  in  eq.  (10)  of  §  34,  viz.: 

r     ^ 
..  L=—\uiVi-unvn], •     (4) 

y 

in  which  u\  and  un  are  the  "' velocities  of  whirl"  of  the  water 
at  entrance  and  exit  respectively;  i.e.,  the  projections  of  the 
absolute  velocities  Wi  and  wn  upon  the  tangents  to  the  two 
wheel-rims. 

Evidently  Ui=wicosa.    As  to  un,  note  that  since  cn  is 
to  be  equal  to  vn  the  triangle  N'C'T'  is  isosceles  and  that 

hence  the  angle  C'N'T'  between  wn  and  vn  is  90°—  -~-;  hence 
un=wn  cos  90°  —  I,  or  un=wn  sin  -.  We  have,  also,  from  the 
same  triangle,  wn  =  2vnsm-;  substitution  of  all  of  which 

values  in  eq.  (4),  with  vi=— — -,  and  vn=—vi,   gives  rise  to- 

Zi  cos  OL  r\ 

the  relation 

L,  or  R'v',  = ~\  1 — ^5 ^ — •  I  (5) 

J  rt  VI  /y»    2i         rm&A    >^/  \v/ 


o      I  o  o 

J   L       FI      cosz  a  _ 
(N.B. — This  is  identical  with  what  would  be  obtained  in 
another  way,   viz.,   by  deducting  the  kinetic  energy  —  •-— 

carried  away  each  second  by  the  water  at  exit,  from  the  kinetic 

Or  Wi2 
energy  -  —  • -5-  arriving  each  second  at  the  entrance  1.     There 

(J  --. 

is  no  change  in  pressure  energy,   nor  in  potential,   between 
entrance  and  exit.) 

If  the  whole  head,  h,  of  the  mill-site  be  considered  as  pro- 
ducing the  entrance  absolute  velocity  Wi  (orwi=V2gh)}  fric- 
tion being  thus  entirely  ignored  in  the  supply-pipe  and  nozzle, 
just  as  it  has  been,  so  far,  in  the  wheel  itself,  eq.  (5)  may  be 
written 

(6) 


cos  a 


78  HYDRAULIC    MOTORS.  §  55. 

It  is  seen  from  eq.  (6)  that  the  theoretical  efficiency 

....     (7) 


cosa 

from  which  it  is  evident  that  not  only  does  a  small  value  for 
d,  but  for  a  as  well,  conduce  to  an  increase  of  efficiency;  though 
a  value  less  than  20°  for  a  is  rarely  used. 

On  substitution  of  the  values  a  =20°,  £  =  15°,  and  rn+ri 
=  1.25  we  obtain  9  =97  per  cent.;  but  in  actual  practice  it 
rarely  rises  over  80  per  cent.,  on  account  of  friction  and  im- 
perfect guidance  of  the  water. 

For  inward-flow  Girard  wheels  the  theory  does  not  differ 
from  the  foregoing,  but  the  angle  d  at  the  exit-point  must 
be  taken  a  little  larger. 

55.  Numerical  Example.  Girard  Impulse  Wheel.  —  With  a 
head  of  &  =  144  ft.  and  a  water-supply  of  Q  =  2  cub.  ft.  per  sec., 
it  is  required  to  design  an  outward-flow  Girard  wheel  with 
parallel  crown-  plates,  taking  a  =25°,  <5  =  20°,  and  the  ratio 
rn-^ri=4-^3.  The  foregoing  theory  will  be  applied,  with  no 
account  of  friction,  at  first,  except  in  the  nozzle.  There  being 
supposed  to  be  no  loss  of  head  between  the  surface  of  head- 
water and  the  jet,  except  in  the  nozzle  itself,  we  have 

wi  =0.95v'2<j^==0.95V'64.4x  144  =  91.4  ft.  per  second. 
The  best  velocity  for  the  inner  rim  will  then  be,  from  eq.  (3), 

wi  91.4 

Vl  =2~^  =  2^0906  =  50'5  ft'  per  Sec' 

(With  friction  considered,  this  might  be  reduced  to  47  or 
48  ft.  per  sec.) 

If  it  be  desired  that  the  wheel  make  240  revs,  per  minute, 
or  4  per  sec.,  we  obtain  a  value  for  n,  the  inner  radius,  by 
writing  4x2^ri=50.5;  obtaining  7*1=  2.01  ft.;  and  hence 
rn  =  (4:3)r1?  --=2.68  ft. 

As  to  e,  the  proper  distance  apart  of  the  two  flat  crown- 
plates,  or  rings,  if  the  "'free  jet  "  at  point  1  (Fig.  37)  is  given 
a  horizontal  thickness  of  Z0=-f  inch,  the  vertical  dimension  of 
its  rectangular  cross-section  will  be  e,  and  we  may  write  Q  = 


•§  56.  GIRARD    IMPULSE    WHEELS.  79 

whence 

o 

->  =4.2  inches. 


While  the  theoretical  efficiency  would  be 

Bin  */2\l  /4  0.174V 

'  =     "13-0906J  :=0'93' 

the  actual  performance  would  probably  be  in  the  neighborhood 
•of  from  75  to  80  per  cent.  On  the  basis  of  75  per  cent,  the 
useful  power  would  be 

L,  =R'vf,  =0.75(^=0.75X2X62.5X144, 
=  13500  ft.-lbs.  per  sec.;  =24.5  H.P. 

If  the  radius  rf  of  the  pulley  (on  same  shaft  as  water-wheel), 
to  whose  circumference  the  resistance  R'  is  to  be  applied,  is 
y  =  1  ft.,  the  velocity  of  a  point  in  that  circumference  would  be 

vf,  =  ~PI,  =777:7X50.5,  =25.2  ft.  per  sec.;  and  therefore  the 

7*1  .Z.U1 

necessary  value  of  Rf  would  be 


56.  Bell-mouthed  Profiles.  —  When  the  distance  between  the 
two  crown-plates  of  an  outward-flow  Girard  wheel  is  the  same 
at  outlet  as  at  entrance  of  the  space  between  two  adjacent 
vanes  a  small  value  of  the  angle  d  may  occasion  too  narrow  a 
passageway    between   the    vanes   at   exit.     If,    however,    the 
crowns  diverge  toward  exit,  making  what  is  called  a  "bell- 
mouthed  "  profile,  the  stream  of  water  becomes  thinner  per- 
pendicularly to  the  vane,  on  account  of  lateral  spreading  along 
the  surface,  and  choking  of  the  passageway  is  prevented. 

Openings  are  frequently  made  in  the  crowns  to  facilitate 
the  escape  of  air,  with  the  same  object  in  view. 

57.  Practical  Construction   of  Girard  Wheels.  —  The    Girard 
wheel  is  a  favorite  type  in  Europe,  some  motors  of  this  kind 
developing  as  much  as  1000  horse-power. 

Several  are  working  at  the  Terni  Steel  Works,  in  Italy, 


80  HYDRAULIC    MOTORS.  §  57. 

from  50  to  1000  H.P.,  under  a  head  of  nearly  600  ft.  One 
of  the  smaller  of  these  is  shown  in  Fig.  38  (on  p.  74)  in  which 
W  is  a  hand-wheel  for  opening  the  gate  in  the  main  supply-pipe, 
P.  The  wheel  revolves  on  a  horizontal  shaft  S  and  is  seen  to 
be  of  outward-flow  and  "  bell-mouthed  "  design. 

The   larger  wheels  at   Terni   are    practically   of   the   same 
general   design.     In   the   case    of   the    800-H.P.    wheel   which 
drives  the  rolling-mill  machinery,  frequent  stopping  and  start- 
ing being  necessary,  a  lateral  pipe  8  ins.  in  diameter  is  provided, 
opening  out  of  the  main  supply-pipe,  whose  diameter  is  24  ins., 
the  gate  of  the  smaller  pipe  being  so  connected  with  the  gates 
admitting  water  to  the  wheel  that  when  the  latter  are  closed 
the  former  is  opened  and  vice  versa.     In  this  way  the  motion 
of  the  water  in  the  main  supply-pipe,  which  is  very  long,  is 
not  entirely  checked  when  the  water  is  shut  off  from  the  wheel, 
but  finds  a  vent  through  the  smaller  pipe;    and  thus  "  water- 
hammer  "  (i.e.,  excessive  rise  of  pressure)  in  the  main  pipe  is 
prevented.     (See  §  125.)     The  outer  diameter  of  this  wheel  is 
9  ft.  5  ins.;    the  inner,  8  ft.  2.4  ins.     The  distance  between 
crowns  at  entrance  is  4.91  ins.;    that  at  exit,  16.14  ins.;    and 
the  quantity  of  water  used  is  Q  =  16  cub.  ft.  per  second,  while 
the  normal  speed  is  200  revs,  per  min. 

In  Fig.  40  is  shown  a  vertical  section,  through  the  axis  of 
shaft  and  also  of  supply-pipe,  of  a  1000-H.P.  Girard  wheel  at 
Vernayaz,  Switzerland;  one  of  six  in  an  electric  power- 
station,  each  of  1000  H.P.  and  working  under  a  head  of  1640  ft. 
The  velocity,  vn,  of  outer  rim  is  normally  184  ft.  per  second. 
The  outer  diameter  is  of  the  wheel  2.150  meters,  or  about 
6.5  ft.;  and  that  of  the  supply-pipe,  0.30  meters.  To  prevent 
too  rapid  ''speeding  up  "  of  the  wheel  when  the  resistance,  Rf, 
or  "load,"  is  diminished,  two  heavy  steel  rings  are  shrunk  on 
the  wheel  on  the  outside  (these  are  seen  in  section  in  Fig.  40), 
and  thus  form  a  fly-wheel.  As  is  evident  from  the  figure,  the 
profile  between  crowns  is  "bell-mouthed." 

Fig.  39  gives  a  cross-section  at  right  angles  to  the  shaft 
and  midway  between  crowns,  and  shows  the  nature  of  the 
nozzle  and  of  the  regulating  apparatus.  Through  action  of 


FIG.  40. 


81 


82  HYDRAULIC    MOTORS.  §  57. 

the  centrifugal  governor-balls  at  C,  and  the  intervening  me- 
chanism, when  the  "load"  on  the  wheel  changes  from  the 
normal,  and  a  slight  change  of  speed  is  thus  brought  about, 
one  edge  of  the  rectangular  opening  which  forms  the  jet  is  caused 
to  move,  and  thus  to  diminish  or  increase  the  thickness  of  the 
jet,  and  thus  vary  the  amount  of  the  working  force  acting  on 
the  wheel. 


CHAPTER  V. 
TURBINES  AND  REACTION  WHEELS. 

58.  *  Reaction  Turbines." — A  turbine  proper,  or  "  reaction 
turbine/'  is  a  hydraulic  motor  consisting  generally  of  two 
•crown-plates  or  shells  (surfaces  of  revolution)  mounted  on  an 
axle,  the  space  between  the  shells  being  divided  by  rigid  curved 
.blades  or  vanes  (" buckets7')  into  numerous  curved  passage- 
ways, or  channels,  distributed  regularly  around  a  circum- 
ference. The  mouths  of  these  channels  receive  water  simul- 
taneously, and  all  around  the  periphery,  from  the  extremities 
of  certain  fixed  guide-channels  and  discharge  it  at  the  turbine- 
channel  exits  either  into  the  atmosphere  or  into  a  space  rilled 
with  water  (whose  internal  pressure  is  frequently  less  than 
that  of  the  atmosphere) . 

The  special  feature  of  the  turbine  as  distinguished  from 
Girard  wheels  is  that  all  of  its  channels  or  passageways 
are  simultaneously  in  action  and  are  completely  filled  with 
water,  flowing  under  pressure.  By  the  proper  design  of  the 
wheel  or  turbine,  and  restriction  of  its  velocity  of  rotation  (as 
accomplished  by  the  imposition  of  a  certain  resistance),  the 
course  of  the  water  is  so  deviated  from  the  path  it  would  take 
if  the  wheel  were  not  present  that  its  absolute  velocity  is  grad- 
ually reduced,  and  its  internal  pressure  brought  to  an  equality 
with  that  of  the  space  into  which  it  is  discharged;  so  that  the 
water  exerts  pressure  or  working  forces  against  the  vanes,  thus 
-enabling  the  turbine  to  maintain  its  uniform  motion  notwith- 
standing the  resistance.  In  steady  operation  the  flow  of  the 
water  is  "  steady/'  or  permanent,  as  already  defined. 

59.  The  Reaction  Wheel,  or  Barker's  Mill.  Theory.  (This 

83 


84 


HYDRAULIC    MOTORS. 


§59.. 


theory  will  now  be  given,  as  preliminary  to  that  of  the  modern 
turbine.) — A  simple  form  of  reaction  wheel  consists  of  a  single 
rigid  casing  secured  transversely  to  a  vertical  axle,  and  is  pro- 
vided with  two  small  orifices  in  the  vertical  sides  of  the  casing, 
each  facing  backwards  as  regards  direction  of  motion,  and 
equidistant  from  the  axle.  (See  Fig.  41.)  A  water-tight  joint 
at  n  prevents  leakage  when  water  is  passing  from  reservoir  W ,. 
by  the  fixed  pipe  m.  .n,  to,  and  through,  the  moving  pipe  and 
casing  n.  .AB.  The  area,  F,  of  each  orifice  is  supposedly  small 


FIG.  41. 

compared  with  the  cross-section  of  the  casing,  AB,  so  that 
the  relative  velocity  of  the  water  in  the  main  body  of  the  latter- 
may  be  considered  to  be  zero.  The  horizontal  plane  of  rota- 
tion of  the  centers  of  the  orifices  is  h  feet  below  the  reservoir 
surface,  and  the  absolute  velocity  of  the  water  at  the  point  o 
in  the  casing  (in  the  axis  of  motion  of  the  latter)  is  so  slight 
that  the  internal  fluid  pressure  there  is  practically  pa  +  hf 
(where  pa  denotes  atmospheric  pressure).  As  above  indicated,, 
the  relative  velocity  c\  of  the  water  at  o  is  to  be  taken  as  zero. 
A  moderate  resistance,  R'  Ibs.,  being  provided  (tension  in  a 
rope  winding  on  drum,  say,  as  shown),  and  the  two  orifices- 
being  opened  (whole  apparatus  originally  full  of  water),  a  flow 


§  59.  REACTION    WHEELS.  85 

begins  and  the  motion  of  water  and  motor  soon  adjusts  itself 
to  some  constant  speed  of  rotation,  the  linear  velocity  of  the 
center  of  each  orifice,  at  distance  rn  from  the  axis,  assuming 
some  value  vn,  corresponding  to  which  a  point  in  the  rope, 

r' 
or  periphery  of  the  drum,  has  a  velocity  v',   =—vn,  where  / 

Tn 

is  the  radius  of  the  drum.  In  other  words,  a  steady  flow  for 
the  water,  and  a  uniform  rotary  speed  for  the  motor,  have  set 
in.  It  is  now  required  to  find  the  proper  value  of  vn  that  the 
useful  power,  R'v',  may  be  a  maximum,  considering  friction 
.at  the  orifice  (only). 

The  absolute  velocity  of  the  jet  of  water  at  exit  B  (in  the 
contracted  vein,  where  the  filaments  have  become  parallel 
.and  are  therefore  under  atmospheric  pressure)  is  evidently 

Wn=Cn-Vn,         .......       (1) 

where  cn  is  its  velocity  relatively  to  the  orifice. 

Noting  that  AB  is  a  uniformly  rotating  pipe,  and  taking  o 
as  an  up-stream  point  where  the  relative  velocity  is  zero  and 
the  pressure  is  pa  +  hj-,  and  B  (jet  in  air)  as  a  down-stream 
point  where  the  relative  velocity  is  cn  and  the  pressure  =pa, 
these  two  points  being  at  the  same  level,  and  considering  the 

Cn2 

one  loss  of  head  h",  =&r~  at  the  orifice,  we  may  apply  Ber- 

*9 

noulli's  Theorem  for  such  a  case  (rotating  casing;  see  eq.  (13a) 
in  §  42)  and  obtain 

Cn2       Pa  Pa  +  hr      V^f-Q          Cn2 

¥+7=     "YT+^T-%--  -  -  •  (2) 

Here  £  is  a  "  coefficient  of  resistance7'  for  the  orifice  and  is 
found  by  experiment  to  have  a  value  of  about  V0.125,  or  J, 
for  the  present  case;  the  orifice  being  in  thin  plate,  or  rounded. 
This  reduces  to 


The  weight  of  water  passing  per  second  in  steady  flow  being 
<??•,  let  us  apply  the  equation  of  "  angular  momentum/'  eq.  (10) 
cf  §  34,  to  this  case,  viz.  : 


86  HYDRAULIC    MOTORS.  §  59. 

Qr 
R'v'  =  —  (uiVi—unvn);  (ft.-lbs.  per  sec.),      .     .     (4) 

Ui  and  un  being  the  projections  of  the  two  absolute  velocities 
Wi  and  wn  (at  entrance  and  exit)  upon  the  "wheel-rim"  veloci- 
ties v\  and  vn.  Now,  in  the  present  case,  at  the  entrance- 
point  o  (see  Fig.  41)  the  absolute  velocity  of  the  water  is  prac- 
tically zero  (large  passageway),  and  the  velocity  of  that  point 
of  the  motor  is  v\  =zero.  At  the  point  of  exit  (jet  in  the  air) 
the  velocity  of.  the  mid-point  of  the  orifice  is  vn  and  the  pro- 
jection of  the  absolute  velocity  upon  the  line  of  vn  is  wn  itself, 
which  is  numerically  equal  to  cn  —  vn;  but  since  this  projection 
points  backward  with  respect  to  the  motion  of  the  wheel  we 
write  it  negative  in  the  substitution;  and  hence  eq.  (4)  re- 
duces to 


Or  Or 

or,  R'v'=^-(cn-Vn)vn,  =-j(cnvn  -vn*).     .     .     .     (6) 

Now  the  efficiency  of  the  motor  is  t]  =R'vf  +Qrh;   hence,  sub- 
stituting from  (6),  and  the  value  of  h  from  (3),  we  have 

2(cnvn-vn2) 

^(l+OCrf-Vn2' 

In  (7)  we  have  )?  as  a  function  of  two  variables,  cn  and  vn> 
but  it  is  of  such  a  character  that  it  can  be  reduced  to  a  function 
of  the  one  variable  x,  if  x  denote  the  ratio  cn'.vn',  that  is,  if 
for  cn  we  write  xvn,  (7)  becomes 


By  obtaining  dy/dx  and  placing  it  equal  to  zero,  we  derive 
=  —  1;   and,  finally,  taking  plus  sign  of  radical, 


1+itc 


as  the  special  value  of  x  that  makes  the  efficiency  a  maximum. 
With  £  =  0.125,  or  f,  we  find,  from  -(9),  z=£,  whence  cn=|vn; 


§  59.  REACTION    WHEELS.  87; 

and  also,  from  (8),  a  value  of  ^=f,  or  G6f  per  cent.,  as  the 
maximum  efficiency. 

For  this  maximum  efficiency  to  be  obtained  it  is  necessary 
that  vn  be  regulated  to  a  value  of  vn  =  \/2gh,  as  obtained  from 
eq.  (3)  when  for  cn  we  write  %vn.  At  this  special  speed  we 
find  also  that  the  maximum  power  for  a  given  Q  is  R'v'  = 
and  that  the  absolute  velocity  at  exit,  =wn,  =cn-vn,  = 

so  that  ^.^  = 

9     z 

That  is  to  say,  of  the  whole  power  of  the  mill-site  (viz., 
Qfh  ft.-lbs.  per  sec.)  two  thirds  is  usefully  employed  in  over- 
coming the  resistance  R' ',  one  ninth  is  carried  away  in  the 
effluent  jet  in  the  kinetic  form,  while  the  remaining  two  ninths 
is  lost  in  friction  at  the  edges  of  the  orifice  (when  the  speed 
is  .regulated  as  above  stated  for  maximum  effect).  In  order 
that  the  whole  available  flow,  Q  cub.  ft.  per  sec.,  may  be  utilized 
at  this  special  speed,  the  aggregate  sectional  area  of  the  two 
jets,  in  the  contracted  vein  where  the  -filaments  are  parallel  and 
relative  velocity  is  cnj  must  have  a  value  of  2F  =  Q  -z-cn. 

If  no  friction  whatever  were  considered,  £  would  be  zero  in 
eq.  (9) ,  giving  x  =  l,  or  cn  =vn,  and  ?)=  unity  from  (8) .  But  this, 
is  impossible  since  from  eq.  (3),  which  gives  cn  =  ^2gh  +  vn2  when 
£  is  zero,  cn  is  always  greater  than  vn.  It  is  evident,  however, 
that  as  greater  and  greater  speed  is  permitted,  the  ratio  cn-^vn 
decreases  towards  a  value  of  unity,  and  since  h  is  constant,, 
may  be  made  to  differ  as  little  as  we  please  from  unity  by  a 
proper  increase  of  vn.  While,  mathematically,  the  efficiency 
would  not  become  unity  except  for  vn=  infinity,  it  would  be 
high  for  values  of  vn  which  are  not  excessive;  e.g.,  for  vn  =  V2gh,. 
vtyh,  and  VSgfi,  we  should  find  y  to  be  0.83,  0.90,  and  0.94, 
respectively.  This  is,  of  course,  for  the  ideal  case  of  no  fric- 
tion. With  great  speeds  of  rotation  fluid  friction  increases 
fast,  as  also  the  resistance  of  the  air  to  the  motion  of  the  motor. 

Weisbach's  experiments  with  a  small  reaction  wheel  some 
3  ft.  between  the  two  orifices  under  a  head  of  h  =  1.3  ft.  con- 
firmed the  above  theory  where  friction  at  the  orifice  has  been 
considered,  with  £  =  0.125. 


88  HYDRAULIC    MOTORS.  §    60. 

In  the  foregoing  theory  it  has  been  virtually  supposed  that 
Q  was  constant  at  all  speeds  of  rotation;  which  implies  a  vary- 
ing size  of  orifice,  since  Q=mFcn,  where  m  is  the  number  of 
orifices  and  F  the  sectional  area  of  the  contracted  vein  of  jet; 
that  is,  a  different  F  would  apply  to  each  different  speed.  If 
the  value  of  F  were  fixed,  Q  would  be  variable,  depending  on 
the  speed;  and  the  outcome  of  the  theory  would  be  different. 
However,  if  a  special  value  of  Q  is  desired  to  be  used  at  any 
particular  speed,  a  proper  size  of  orifice  is  easily  computed 
to  secure  this  result,  since  eq.  (6)  is  independent  of  the  size 
of  orifice  so  long  as  the  latter  is  small  compared  with  the  sec- 
tional area  of  the  casing. 

60.  Reaction  Wheel.  Theoretical  Points.  —  The  reaction 
wheel,  though  now  obsolete,  presents  some  interesting  theo- 
retical features.  The  expression  for  the  useful  power,  R'v'  , 
as  already  derived,  and  stated  in  eq.  (6),  may  be  transformed 
as  follows. 

It  may  be  written  thus: 

Or 

R^=f-[2cnvn-vn2-vn2l      ....     (10) 
u 

We  may  then,  in  the  bracket,  add  the  quantity  2gh+vn2  —  £cn2, 
.and  subtract  its  equal,  cn2  (see  eq.  (3))  ;  whence 
Or 

vJ-tcJ-W-tenVn+V,?)  -Vn2]',         (11) 


which,  since  cn  —  vn  =  wnj  reduces  to 


which  is  the  same  expression  for  the  power  as  might  have  been 
derived  by  deducting  from  the  whole  theoretical  power,  Qrh, 
of  the  mill-site,  the  kinetic  energy  carried  away  each  second 
by  the  water  in  the  effluent  jets  by  virtue  of  its  absolute  veloc- 
ity wn  and  the  power  lost  in  friction  at  the  orifice  (i.e.,  the 
product  of  the  Ibs.  of  water  flowing  per  second  by  the  "friction- 
head/'  or  "loss  of  head/7  due  to  the  passage  through  the  orifice. 
61.  Working  Forces  in  Barker's  Mill.  —  Another  interesting 
matter  is  the  nature  and  position  of  the  actual  working  forces 


•§  61. 


REACTION   WHEELS. 


89 


B 


ft)' 

I 


-or  pressures  which  are  exerted  on  the  inside  wall  of  the  casing 
during  steady  operation,  enabling  the  motor  to  keep  up  the 
motion  uniformly  notwithstanding  the  resistance  Rf. 

Fig.  42  shows  a  horizontal  section  of  the  casing  of  Fig.  41, 
but  it  is  now  supposed  to  have  vertical  side  walls;  the  two 
orifices  indicated  being  under 
like  conditions.  The  rotation  is 
counter-clockwise,  with  a  con- 
stant angular  velocity  n>,  so  that 
vn  =  asr= linear  velocity  of  orifice. 
B  is  the  jet,  in  the  atmosphere, 
having,  at  the  contracted  sec- 
tion where  the  sectional  area  is 
F,  a  velocity  cn  relatively  to  the 
orifice.  The  casing  is  so  wide 
that  at  Bf,  in  the  interior,  just 
inside  from  the  orifice,  the  rela- 
tive velocity  of  the  water  is 
practically  zero,  while  its  abso- 
lute velocity  at  B'  is  vn,  =that 
of  the  orifice  itself.  At  C  the 
excess  of  pressure  of  the  water 
against  the  vertical  wall  of  cas- 
ing over  that  on  the  corresponding  portion  of  wall  from  which 
the  orifice  is  cut  out,  or  "reaction"  of  the  jet  on  the  casing, 
is  a  force  P  whose  value,  according  to  p.  800  of  M.  of  E.,  is 
P  =  2<j)2Fhr,  friction  at  the  orifice  being  considered.  But  the 
h  of  this  expression  was  equal  to  the  (v2  -^2gr)  of  p.  800,  v  being 
there  the  velocity  of  the  jet  relatively  to  the  vessel  (=cn  in 
our  present  notation),  and  <f>  the  " coefficient  of  velocity," 
which  is  the  same  as  1  -f-Vl  +  £  (see  p.  706,  M.  of  E.,  and  eq.  (2) 
of  the  foregoing) .  That  is,  at  C  we  have  a  working  force  of 


(13) 


and  a  similar,  equal,  force  in  connection  with  the  other  orifice. 
At  first  sight  it  might  seem  that  all  other  horizontal  pres- 


FIG.  42. 


90  HYDRAULIC    MOTORS.  §  GL. 

sures  on  the  inside  walls  of  the  casing  were  balanced;  but 
since  the  water  is  being  caused  to  travel  out  from  the  center 
o  toward  the  position  B',  acquiring  an  increasing  absolute 
velocity  as  it  proceeds,  the  casing  has  to  act  as  a  centrifugal 
pump  to  that  extent  and  consequently  must  encounter  re- 
sisting forces  due  to  this  cause.  This  resistance  consists  in 
the  fact  that  the  water  pressures  along  GH  (and  G"H"),  the 
rear  vertical  walls  of  the  casing,  are  greater  than  those  along 
DE  (and  D"E"),  the  front  walls.  These  pressures  .constitute 
a  couple  in  a  horizontal  plane  whose  moment,  AT,  may  be 
found  from  eq.  (9o)  of  §  34,  i.e., 

Or 
M'=—  [uiri-unrn]  .  -  -  (ft.-lbs.).    .     .     .  (13a) 

c/ 

Here  HI  and  un  are  the  tangential  components  of  the  abso- 
lute velocities  of  the  water  at  the  two  points  in  the  rotating 
casing  between  which  the  forces  in  question  act,  viz.,  o  and 
B'  in  Fig.  42,  and  r*i  and  rn  the  two  corresponding  radii.  Evi- 
dently ui  is  zero  and  un  =  vn;  therefore 


(14) 


The  negative  sign  shows  that  this  couple  tends  to  retard 
the  motion  of  the  casing  instead  of  furnishing  working  forces. 

We  are  now  able  to  formulate  the  net  power  (ft.-lbs.  per 
sec.)  due  to  the  two  working  forces  P  and  P  and  the  resisting 
forces  constituting  the  couple  whose  moment  is  Mf  ;  remember- 
ing that  the  work  done  per  second  by  the  couple  is  the  product 
of  its  moment  by  the  angular  velocity  a>  of  the  casing,  i.e., 


*-"J 

Substituting  vn  for  wrn  and,  for  P,  its  value  as  found  in 
eq.  (13),  we  have 

T>f    t         J~l~  /  2\  f~\f\\ 

ft.-lbs.  per  second,  as  before  obtained;  see  eq.  (6). 

The  foregoing  applies  equally  well  to  a  casing  of  any  form 
(orifice  small,  however)  when  in  place  of  the  pressures  on  the 


§62. 


TURBINES. 


91 


vertical  sides  of  the  present  form  we  substitute  the  compo- 
nents, in  plane  of  rotation  and  tangent  to  motion)  of  the  actual 
pressures  on  interior  walls. 

62.  Development  of  the  Turbine. — Barker's  Mill  was  im- 
proved by  Whitelaw  and  given  a  form  resembling  that  shown 
in  Fig.  43,  called  the  Scotch  turbine;  furnished  with  three 
orifices,  which  were  made  adjustable  in  size  by  movable  flaps, 
to  provide  regulation  of  the  quantity  of  water  used  and  power 
developed. 

A      nc     A 


FIG.  43. 


FIG.  44. 


We  next  find  in  Combe's  turbine  (Fig.  44)  many  jets,  occupy- 
ing the  entire  circumference,  guided  between  vanes,  or  blades, 
fixed  in  a  ring  attached  to  a  shaft,  the  water  being  supplied 
from  underneath  through  a  fixed  pipe  or  tube.  No  interior 
fixed  guides  were  provided  to  direct  the  water  at  any  special 
angle  upon  the  moving  vanes.  Fig.  44  shows  a  vertical  sec- 
tion of  the  wheel  and  vertical  shaft,  viz.,  BACAB;  and  supply- 
pipe  DD;  also  a  horizontal  section,  H,  of  one  half  of  the  wheel. 
Passing  from  the  fixed  pipe  DD  outwardly  through  the  wheel, 
the  water  completely  fills  the  passages  of  the  latter  and  is  dis- 
charged at  the  outer  rim,  around  the  entire  circumference, 
with  a  relatively  small  absolute  velocity  into  the  atmosphere. 
The  Cadiat  turbine  was  practically  the  same  as  Combe's,  but 
the  supply- pipe  was  placed  above. 

ID  1826  the  French  engineer  Fourneyron  improved  the 
•Cadiat  turbine  by  placing  fixed  guide-blades  just  inside  the 
wheel-ring  around  the  entire  circumference,  by  means  of  which 
the  water  received  a  forward  direction  of  motion  before  enter- 


H 


HORIZONTAL 
SECTION 

N-A-X 


GUIDES 


w 


WHEEL, 
«...         or 
"RUNNER" 


FIG.  45. 


92 


§  63.  TURBINES.  93 

ing  the  channels  of  the  moving  turbine.  This  rendered  attain- 
able a  very  low  value  of  the  absolute  velocity  of  the  water 
at  exit  from  the  outer  rim  of  the  wheel-ring.  Also,  the  wheel 
being  operated  under  water,  the  complete  filling  of  the  wheel- 
channels  was  insured  when  properly  designed.  This  was  the 
first  modern  turbine:  a  motor  which,  as  varied  and  improved 
by  Fontaine,  Henschel,  Jonval,  and  others  in  Europe,  and  by 
Boyden  and  Francis  and  their  successors  in  America,  has  grown 
in  popular  favor  and,  together  with  the  impulse  wheels  already 
described,  has  almost  entirely  supplanted  the  old  forms  of 
vertical  water-wheels  so  long  considered  as  giving  the  highest 
efficiency. 

It  is  the  peculiarity  of  the  turbine  proper  (or  "reaction 
turbine,"  as  distinguished  from  a  Girard  impulse  wheel  or 
"Girard  turbine")  that  the  power  to  be  transmitted  to  the 
wheel  by  the  water  is  present  at  entrance  partly  in  the  form 
of  pressure  energy  and  partly  in  that  of  kinetic;  since  the 
pressure  of  the  water  at  entrance  is  usually  above  that  of  the 
atmosphere. 

63.  Description  of  a  Simple  Fourneyron  Turbine. — Fig.  45 
shows  in  the  upper  part  a  vertical,  and  in  the  lower  part  a 
horizontal,  section  of  a  simple  design  of  a  turbine  of  the  Four- 
neyron type  (or  "  outward-flow,  radial  turbine  ")•  A  case  has- 
been  chosen  of  a  "low-pressure"  turbine,  or  one  for  which  no 
long  supply-pipe  or  penstock  is  necessary,  the  turbine  being 
placed  at  the  bottom  of  an  open  wheel-pit. 

The  water  from  the  head-bay  or  head-water,  H,  descends- 
slowly  through  the  tube,  or  short  penstock,  PP,  which  is  firmly 
supported  and  is  provided  with  a  prolongation,  CC,  or  cylin- 
drical gate,  movable  vertically  and  having  rounded  edges  on, 
its  lower  periphery.  This  lower  edge  is  also  slotted  to  receive 
the  curved  stationary  guides  which  are  shown  (in  'the  hori- 
zontal section)  at  G  and  which  are  rigidly  attached  to  the 
fixed  plate  c.  .  c.  This  plate  is  supported  from  above  by  means 
of  a  pipe  enclosing  the  shaft  of  the  turbine  and  serves  also  to- 
protect  the  lower  shell  DSD  of  the  turbine  from  the  pressure 
of  the  water  in  space  CG. 


ABSOLUTE  PATH     Of 

WATER     ;s 

G-t-N 


FIG.  48. 


94 


§  63.  FOURNEYRON   TURBINE.  95 

The  turbine  itself  and  its  shaft  are  shown  in  vertical  sec- 
tion by  solid  black  shading;  viz.,  EDKDE.  EE  and  DD  are 
the  two  crowns,  or  horizontal  rings,  between  which  are  inserted 
the  curved  vertical  vanes  shown  in  outline  in  the  horizontal 
section  at  W.  The  lower  shell  of  the  turbine  provides  for  the 
rigid  connection  of  the  turbine  proper  (or  crowns  and  vanes) 
with  the  shaft,  and  may  be  lightened  by  perforations.  The 
turbine  illustrated  in  Figs.  17,  18,  and  19  (opp.  p.  42)  is  prac- 
tically of  this  design.  The  resistance  R' ',  which  the  wheel  is 
overcoming,  is  shown  in  Fig.  45,  as  acting  at  edge  of  pulley  M} 
keyed  on  shaft  of  wheel.  The  velocity  of  the  edge  of  the  pulley 
is  v'  ft.  per  sec. 

Fourneyron  placed  a  number  of  horizontal  partitions  between 
the  crowns,  thus  dividing  the  turbine  into  several  stories,  for 
the  purpose  of  preventing  in  some  degree  the  loss  of  head, 
and  consequent  loss  of  power,  resulting  from  the  sudden  en- 
largement of  passageway  which  would  occur  when  the  turbine 
is  operating  at  "'part  gate/'  if  this  device  were  not  adopted. 
In  turbine  parlance,  "full  gate/7  or  "whole  gate/'  refers  to  the 
fact  that  the  spaces  between  the  fixed  guides,  G,  are  fully 
open,  the  gate  being  then  fully  drawn  up,  as  in  Fig.  45;  its  lower 
being  even  with  the  upper  crown.  In  Fig.  46  is  shown 


FIG.  46. 

a  section  of  a  Fourneyron  turbine  furnished  with  horizontal 
partitions  of  the  kind  mentioned.  When  the  lower  edge  of 
the  gate  is  even  with  one  of  these  partitions  only  those  portions 
of  the  channels  which  are  below  this  partition  are  in  action, 


96  HYDRAULIC    MOTORS.  §  64.. 

and  the  efficiency  of  the  turbine,  when  thus  working  at  "'part 
gate  "  and  using  less  than  the  usual  .quantity  of  water,  is  not 
materially  changed. 

64.  Notation  for  Theory  of  Fourneyron  Turbine. — In  Fig.  47 
is  represented  a  portion  of  the  turbine,  and  corresponding 
guides  and  guide-channels,  in  horizontal  section.  The  turbine 
channels  mn,  etc.,  are  so  many  closed  pipes,  supposed  com- 
pletely filled  by  the  water  when  the  turbine  is  in  operation. 
Let  Fn=the  sum  of  all  the  sectional  areas  like  nd  of  the 
turbine  channels  at  the  outer  circumference;  and  F0  the  sum 
of  all  those  like  md0  between  the  stationary  guides,  where 
the  water  is  just  leaving  them  to  enter  the  turbine  or  wheel. 

Let  wi  denote  the  absolute  velocity  of  the  water  leaving 
the  guides  at  point  1,  Fig.  47  (and  at  A  in  Fig.  45).  Also, 
in  Figs.  47  and  48,  let  wn  denote  the  absolute  velocity  of  the 
water  leaving  the  wheel  at  the  exit-rim,  A7,  being  represented 
in  amount  and  position  by  the  diagonal  of  the  parallelogram 
formed  on  cn,  the  relative  velocity  at  A,  and  vn,  the  velocity 
of  the  outer  rim  of  the  wheel  itself. 

Similarly,  at  point  1,  the  absolute  velocity,  wi,  of  the  water 
entering  a  wheel-channel  is  the  diagonal  of  a  parallelogram 
formed  on  its  relative  velocity  at  that  point  and  the  velocity, 
vi.  of  this  inner  rim  of  the  wheel.  Note  that  in  each  case  the 
diagonal  meant  is  the  one  which  springs  from  the  same  corner 
as  the  c  and  the  v.  (For  relative  and  absolute  velocity,  see 
p.  89,  M.  of  E.) 

If  the  wheel  is  run  at  the  proper  speed  and  the  angle  ft 
has  been  given  a  corresponding  suitable  value,  such  that  the 
tangent  to  the  vane  curve  at  1  coincides  in  position  with  the 
relative  velocity  c\  (velocity  of  the  water  leaving  the  guide 
extremities  relatively  to  the  point  1  of  the  inner  wheel-rim), 
there  will  be  no  "elbow"  or  sharp  turn  in  the  absolute  path 
of  the  water  as  it  enters  the  wheel,  but  that  path  will  be  a 
smooth  curve  throughout  its  whole  extent.  See  curve  G.  .  1.  .A" 
in  Fig.  47.  In  this  way,  impact  or  "shock  "  at  entrance  is 
avoided  and  the  corresponding  loss  of  energy  due  to  the  internal 
friction  of  the  water. 


§65. 


FOURNEYRON    TURBINE. 


97 


This  relation  being  stipulated,  it  follows  that  the  absolute 
velocity  of  the  water  just  entering  a  wheel-channel  at  1  is  prac- 
tically the  same  as  the  absolute  velocity  that  it  has  on  leaving 
the  guides;  both  being  therefore  designated  by  w\. 

Fig.  49  shows  by  a  vertical  section  the  notation  used  for 
vertical  heights.  That  from  the  surface  of  head-water  to  that 


H 


FIG.  49. 


of  tail-water,  =h;  while  the 
heights  of  these  surfaces 
above  the  horizontal  plane 
passed  through  a  point  of 
turbine  half-way  between 
the  two  crowns  are  hi  and 
hn  respectively.  The  radii 
of  the  inner  and  outer  edges 
of  the. wheel  are  r\  and  rn 
respectively;  see  Fig.  48. 
The  height  of  wheel,  or  verti- 
cal distance  between  crowns, 
is  e;  the  same  in  this  case 
both  at  entrance  and  exit  of 
a  wheel -channel.  The  mean- 
ing of  the  angles  a,  /?,  /*,  and 
d  is  evident  in  Fig.  48.  Q  denotes  the  number  of  cub.  ft.  of 
water  used  per  sec.,  in  steady  flow.  Let  pi  be  the  internal 
pressure  of  the  water  at  entrance  of  the  wheel;  and  pn  that 
at  exit  from  the  wheel,  i.e.,  at  N. 

65.  Theory  of  the  Fourneyron  Turbine.  Friction  Disre- 
garded.— The  quantities  .Q,  hi,  hn,  ?,  TI,  rn,  OL,  and  d,  being 
given;  it  is  required  to  determine  the  "best"  value  for  the 
velocity  vn  of  outer  wheel-rim  (i.e.,  inducing  the  highest  effi- 
ciency) ;  and  the  proper  height,  e,  between  crowns  that  the 
whole  available  rate  of  flow,  Q,  may  be  used.  We  shall  find 
that  in  the  relations  to  be  written  out  nine  unknown  quanti- 
ties are  involved,  viz.,  vi,  vn,  Wi,  wn,  e,  c\,  cn,  pi,  and  pn,  and 
it  is  evident  that  for  a  complete  solution  nine  independent 
and  simultaneous  equations  will  be  needed.  For  the  present 
all  friction  will  be  disregarded  and  the  simple  design  already 


98  HYDRAULIC    MOTORS.  §  65. 

shown  in  Figs.  45  to  49  inclusive  will  be  the  one  treated.     We 
suppose  the   cylindrical  gate  raised  to  its  full  height   ("'full 
gate  "). 

The  necessary  equations  are  the  following: 
From  the  parallelogram  of  velocities  at  entrance  or  point  1  : 
Cl2=wi2+vi2  —  2wiVi  cosa  ......     (1) 

Similarly,  from  the  parallelogram  of  velocities  at  exit,  or  N, 
wn2  =  cn2  +  vn2  -  2cnvn  cos  d  ......     (2) 

Thirdly,  applying  Bernoulli's  Theorem  for  a  stationary  rigid 
pipe  and  steady  flow  of  water  (see  p.  654,  M.  of  E.)  to  the 
surface  of  the  head-water,  as  up-stream  position,  and  the  point 
1,  where  the  water  leaves  the  guides,  as  down-stream  position, 
we  have 


(where  b  is  the  height  of  the  water  barometer). 

In  its  progress  through  a  wheel-channel  from  1  to  N  the 
water  is  flowing  with  steady  flow  through  a  closed  pipe  rotating 
uniformly  in  a  horizontal  plane  and  we  may  therefore  apply 
Bernoulli's  Theorem  for  Steady  Flow  in  a  (uniformly)  Rotating 
Casing  to  this  part  of  the  path  of  the  water;  hence  (see  eq.  (13), 
'§41) 


0    r      9    r         g 

Since  the  kinetic  energy  carried  away  per  second  by  the 

QY  Wn2 
water  at  exit  is  -—•-—,  and  this  may  be  made  small  by  mak- 

9     ^ 

ing  vn±=cn  in  the  parallelogram  of  velocities  at  N  (in  connection 
with  a  small  value  for  the  angle  d)  (see  also  §  53),  we  shall 
•write  vn  =  cn  .........  (5) 

The  aggregate  sectional  area,  Fn,  of  the  wheel-  passages  at 
'£xit  may  be  expressed  thus:  The  area  of  cross-section  of  any 
'one  channel,  taken  at  right_angles_to  the  vane,  at  exit,  is  (see 
'IJlg.  "47)  F'=eXnd;  but  nd  is  =  nn'-sin  d,  and  hence  F'  = 
'nn'-e-sin  d;  but  the  sum  of  all  the  short  linear  arcs  like  nn' 


;§  66.       THEORY  OF  THE  FOURNEYRON  TURBINE.         99 

making  up  the  entire  outer  periphery  of  the  turbine  (if  we 
neglect  the  thickness  of  the  vanes)  is  2?rrn,  whence  it  readily 
follows  that  Fn  =  2xrne  sin  d.  Similarly,  we  have  Fo=2nrie  sin  a. 
'ButQ  =  Fncn,  and  also  =  F0wi;  therefore  we  have 

[2nrie'Sma]wi=[2xrne'Sm  d]cn,    ....     (6) 

as  also 

Q=[27rrwe-sintf]cn  .......     (7) 

Since  Vi=ajri  and  vn  =  wrn  (w  being  the  angular  velocity  of 
wheel),  it  follows  that 

vi+vn=ri+rn  .......     (8) 

Also  (see  below)  pn  =  rhn+pa  .......  '(9) 

66.  Combination  of  Foregoing  Equations.  —  Since  the  water 
is  supposed  to  leave  the  wheel  at  N  in  parallel  filaments,  the 
outer  of  these  filaments  being  subjected  to  the  hydrostatic 
pressure  rhn  +  pa  (where  pa  is  the  pressure  of  the  atmosphere) 
from  the  surrounding  still  water  in  the  receiving  pool  or  tail- 
water,  the  internal  pressure  of  the  water  at  this  place  may 

be  taken  as  pn  =  rhn  +  pa;  i.e.,  —=hn  +  b  (where  6=—  is  the 
height  of  the  water  barometer).  This  value  being  substituted 
in  eq.  (4),  as  also  the  value  of  —  obtained  from  (3),  eq.  (4) 
becomes 

Cn2       Cl2       t>n2-t>l»  '  W? 

~    =  ~~~    *     n  '    *   ' 


This  last  equation,  on  substitution  of  the  value  of  c\2  from 
•(I),  reduces  to 

2w1vlcosa+cn2-Vn2=2g(hi-hn).      .     .     .     (11) 

But  hi—hn=h;  and,  from  (5),  cn  =  vn,  so  that  (11)  becomes 
WiVicosa=gh  .......     (12) 

Now,  from  eq.  (6),  w\  =  (cnrn  sin  d)  +  (ri  sin  a)  ;  and,  from  (8), 
•Vi=riVn-*-rn',  also  cn=vn  from  (5);  therefore  (12)  becomes 

Velocity  of  outer  rim  for  a  max.  efficiency  =  vn  =  \]  —  =  —  T—  .     (13) 


100  HYDRAULIC    MOTORS.  §  67. 

As  will  be  seen  later,  this  value  may  be  reduced  by  8  per 
cent,  of  itself  to  allow  for  friction,  and  the  resulting  reduced 
Value  used  in  eq.  (6)  for  the  determination  of  Wi,  the  relation 
Jn  =  ^n  being  practically  independent  of  the  consideration  of 
friction.  We  are  then  in  a  position  to  find  the  angle  /?  at 
entrance,  the  angle  a  being  given  and  the  values  of  w^  and  v\,. 
=  (ri-t-rn)vn,  being  now  available. 

This  angle  /?  determines  the  position  which  the  vane- tangent 
at  entrance  should  have  to  avoid  impact  or  "shock"  at  that 
point;  i.e.,  the  vane- tangent  at  1  should  follow  the  direction 
of  the  relative  velocity  cn.  The  vane- tangent  at  exit,  N,  must 
make  the  given  angle  d  with  a  tangent  to  the  outer  wheel- 
circumference  at  that  point.  The  form  of  curve  to  be  given 
to  the  vane  between  points  1  and  N  is  theoretically  imma- 
terial, so  long  as  the  curvature  is  smooth.  Two  circular  arcs 
may  be  used,  the  radius  of  the  part  near  1  being  about  one 
half  of  that  of  the  other  part.  To  a  guide^blade  is  generally 
given  the  form  of  a  single  circular  arc. 

67.  Shorter  Proof  of  Foregoing  Eq.  (12).  (See  Figs.  46-49 
inclusive.) — There  being  no  loss  of  energy  considered  to  take 
place  between  the  surface,  H,  of  head-water  and  the  entrance,. 
1,  of  the  wheel-channels,  and  also  no  loss  due  to  friction  in 
those  passages  themselves,  the  difference  between  the  aggre- 
gate energy  (of  the  weight  Qy  flowing  per  second)  of  the  three- 
kinds  (see  §  9)  at  that  upper  surface  and  that  at  the  point,  N, 
of  exit  from  the  wheel-channels,  should  represent  the  power, 
L,  (ft.-lbs.  per  sec.,)  exerted  by  the  water  on  the  turbine. 

The  horizontal  plane  through  N  will  be  taken  at  datum-plane 
for  the  potential  energy.  At  H  the  weight  Qj-  has  Q^rhi  ft.-lbs. 
of  potential  energy,  zero  of  kinetic  energy,  and  Qfb  of  pressure 
energy;  while  at  N  its  potential  energy  is  zero,  kinetic  energy 

2r.^L,   pressure  energy -QrV'  =<2K&+W,  =  Qrt&+fti  -A). 
g     *  T 

Subtracting  the  sum  of  the  latter  three  items  from  that  of  the 
former  three,  we  have 


§  67.  THEORY    OF    THE    FOURNEYROX    TURBINE.  101 

ft.-lbs.  per  second;  and  this  should  be  equal  to  the  wo»k  done 
per  second  by  the  "  equivalent  couple "  (see  eq.  (7),  §34) 
on  the  turbine,  an  expression  for  which  work  per  second  we  have 
in  the  "'angular  momentum  "  equation,  eq.  (10)  of  §  34,  viz., 

L=—(u1v1-unvn) (15) 

t7 

(and  this  relation  holds  true,  also,  when  friction  is  considered). 
But  ui,  the  projection  of  wi  on  the  inner  whe^l-tangent, 
is  wicosa;    and   similarly,    at   the   outer  rim,   un*=wn  cos  p. 
Hence  (15)  becomes 

Qr 

L  = — (wivi  cos  a  -[wn  cos  p]vn) ....     (16) 

*J 

The  right-hand  members  of  eqs.  (14)  and  (16)  being  equated, 
there  results 

w  2 
w^i  cos  a-(wncosfjL)vn=gh — |-.    .     .     .     (17) 

Let  now  the  parallelogram  of  velocities  at  the  exit-point  N 
be  reproduced  in  Fig.  50  (the  direction  of  rotation  (clockwise) 
of  the  turbine  is  contrary  to  that  of  previous  figures).  The 
condition  that  cn  =  vn  for  best  effect  has  been  introduced  into 
this  figure  by  making  it  a 
rhombus,  with  side  DN 
equal  to  side  NE.  The 
diagonals  bisect  each  other 
at  right  angles,  and  BN 
represents  \wn.  Hence 
the  intersection,  B,  of  the 
diagonals,  lies  in  the  cir- 
cumference of  the  semi- 
circle described  on  NE  (or  vn)  as  a  diameter.  Hence,  if  BO 
be  drawn  perpendicular  to  NE,  BN  is  a  mean  proportional 
between  NO  and  NE-,  or^N2^NO^NE',  i.e., 

OOS/A  Wn2 

2 )Vn'    °T}     (v«coB/d*-"Tjp     •    (18) 

Substituting  from  (18)  in  (17)  we  obtain 

gh, (19) 


102  HYDRAULIC    MOTORS.  §  68. 

as  holding  good  when  the  turbine  (frictionless)  is  running  witk 
speed  of  maximum  efficiency,  the  same  as  eq.  (12). 

68.  Note.  —  It  is  to  be  noted  that  this  same  demonstration 
for  eq.   (19),  with  same  result,  will  hold  good  whatever  the 
positions  of  the  planes  of  the  parallelograms  of  velocities  at 
points  1  and  N,  entrance  and  exit,  of  the  turbine;   since  the 
projections  u\  and  un  would  always  be  in  the  same  lines  as  the 
wheel-rim  velocities,  vi  and  vn,  respectively.     Eq.  (19)  holds, 
therefore,  for  all  kinds  of  turbines. 

69.  Theoretical  Efficiency  of  the  Fourneyron  Turbine.  —  It  is 
evident  that  the  value  of  hn  or  depth  of  the  wheel  below  the 
surface  of  the  tail-water  is  immaterial,  since  hn  is  offset  by  ari 
equal  portion  of  the  height  hi;    hence  we  may  formulate  the 
power  transferred  to  the  wheel  by  the  water  (on  the  present 
basis;   friction  disregarded;    i.e.,  no  loss  of  head  either  in  the 
penstock  between  surface  of  head-water  and  entrance  of  tur- 
bine, nor  in  the  turbine  itself)  by  supposing  hn  to  be  zero. 
That  is,  this  power,  L,   (ft.-lbs.  per  second,)  equals  the  whole 
theoretic   power  of  the  mill-site  Q-fh  less  the  kinetic  energy 
carried  away  per  second  by  the  water  leaving  the  wheel  at  N,  or 

OY  nr  2 

L,  =R'v',  =Qrh-^-^  .....    (18o) 

9      ^ 

Since  the  condition  that  cn  =  vn  makes  the  parallelogram  of 

velocities  at  N  consist  of  two  isosceles  triangles  (see  also  Fig.  50) 
« 

we  have  Wn  =  2vnsm—,  in  which  if  the  value  of  vn  for  best 

effect  as  derived  in  eq.  (13)  be  substituted,  and  the  result  so 
obtained  for  wn  placed  in  (18«),  we  have 


2  tan  a  sin2— 

L, 


J 


In  this  case  the  efficiency,  y,  =Rfv'  -^Qfh,  whence 


2  tan  a  sin2— 

-KIT-  ......  (20) 


§  70.  THEORY    OF   THE    FOURNEYRON   TURBINE.  103 

From  the  details  of  this  expression  we  gather  that  the  smaller 
the  angles  a  and  d  can  be  made,  the  greater  the  efficiency. 
In  practice  a  is  taken  from  20°  to  30°;  and  d  from  15°  to  20°. 

With  the  values  of  a  =25°  and  d  =  15°  we  obtain  )?=0.92 
from  eq.  (20);  but  in  actual  practice  this  figure  is  reduced  to 
80  per  cent,  or  less  (unless  in  exceptional  cases)  on  account  of 
fluid  friction,  axle  friction,  and  imperfect  guidance  of  the  water. 
75  per  cent,  is  a  fairly  good  performance. 

70.  Numerical  Example.  Fourneyron  Turbine.  —  Given  h  = 
60  ft.  and  the  available  water-supply  Q  =  150  cub.  ft.  per  sec., 
and  assuming  radii  of  ri=2  and  rn  =  2.5  ft.;  with  angles  a  and 
d,  20°  and  15°,  respectively;  it  is  required  to  design  a  Fourneyron 
turbine  having  parallel  crowns,  etc.,  as  in  Fig.  46;  i.e.,  to  find 
the  proper  value  of  the  outer-rim  velocity,  vn,  for  best  effect, 
that  of  the  angle  /?  for  the  vane  tangent  at  entrance  and  the 
proper  distance,  e,  between  crowns,  that  all  the  water  avail- 
able may  be  used  (at  full  gate). 

Up  to  this  point  the  effect  of  fluid  friction  has  not  been 
represented  in  any  of  the  formula,  but  a  fair  allowance  for 
it  may  be  made  (see  §  71)  by  deducting  8  per  cent,  of  itself 
from  the  value  of  vn  for  best  effect,  as  given  by  eq.  (13)  in  §  66; 
i.e.,  with  tan  20°  =  0.364  and  sin  15°  =0.259,  we  have 

32.2X  60X0.364 

=48  ft.  p.  sec. 


With  rn  =  2.5  ft.,  this  means  that  the  wheel  should  be  run 

at  an  angular  velocity  of  w=—  =  19.2  radians  per  sec.,  or  at 

z.o 

(19.2  4-2;r)  X  60  =  183  revs,  per  minute. 

(Should  it  be  wished  to  run  the  wheel  at  a  different  angular 
velocity,  a  different  value  of  the  radius  rn  could  be  selected, 
so  long  as  the  value  of  the  linear  velocity  vn  of  the  outer  rim 
is  kept  unchanged.) 

Since  cn=vn  we  have,  from  eq.  (6), 

Wi  (27rrie  sin  a)  =vn(2nrne  sin  d) 
(which  holds  good  whether  friction  be  considered  or  not)  ;  and 


104 


HYDRAULIC    MOTORS. 


70. 


hence  for  the  absolute  velocity  of  the  water  leaving  the  guides 
vnrn  sin  d    48X2.5  0.259 


sin 


also, 


+rn  =  (2  -5-2.5)48 =38.4  ft.  p.  sec. 


To  determine  the  vane-tangent  angle,  /?,  at  point  1,  i.e., 
the  position  of  the  relative  velocity  ci,  v\  and  w\  being  now 
known   and   angle  ya  being  given,  we  have 
only  to  solve  the  triangle  ABC  in  the  paral- 
lelogram of  velocities  concerned;  see  Fig.  51. 
Here  we  have  two  sides  (wi}  vi)  and  the  in- 
cluded angle  (a);   the  other  two  angles  being 
C  and  6,  (0  =  180°-/?.)     Hence 
(wi-vi)  tan 


7Xtan8° 


=0.473; 


FIG.  51. 


Hence 


and       /?, 


83.8 
•'•  i(0-C)=25°19'. 

(80° +  [25°  19'])  =  105° 
180° -0,=  74°  41'. 


19'; 


(N.B.  Another  method  of  solving  the  triangle  and  finding 
£  is  illustrated  in  §  94  and  Fig.  77.) 

To  find  e,  the  distance  between  crowns,  i.e.,  the  common 
height  of  all  parts  of  all  wheel-passages  (at  full  gate),  we  have 
from  eq.  (7)  Q=2nrne(sm  d)cn,  and  again  'write  vn  for  cn 
and  obtain  e  =  150 +  (2nX 2.5x0.259x48)  -0.768  feet. 

As  no  account  has  been  taken  of  the  thickness  of  the  guides 
or  vanes,  this  value  for  e  would  need  to  be  increased  some- 
what, perhaps  10  per  cent,  in  some  cases  (see  §  91  for  further 
details  on  this  point). 

As  to  the  horse-power  to  be  expected  from  the  turbine 
when  run  at  the  proper  speed  deduced  above  (183  revs,  per 
minute),  assuming  an  efficiency  of  75  per  cent,  (not  an  extrava- 
gant figure),  we  have  for  the  useful  power 


;§  71.  THE    FOURNEYRON    TURBINE.      EXAMPLE.  105 


R'v'  =0.75<3?^  =  0.75  X  150X62.5X60, 

=  421,500  ft.-lbs.  per  sec.;  =  766  H.P.; 

•equivalent  to  the  continuous  raising  (vertically)  of  a  weight 
of  #',=42,150  lbs.;  at  a  uniform  speed  of  i>'  =  10  ft.  per  sec. 

71.  Theory  of  the  Fourneyron  Turbine,  when  Friction  is 
Considered.  —  Let  us  now  consider  that  in  the  steady  flow 
between  the  head-  water  surface  H  and  the  outlet,  A,  of  the 

U>i2 

guides  (Fig.  45)  a  loss  of  head  occurs  of  the  form  £<r«-,  and 

\j 

introduce  it  into  eq.  (3)  of  §  65;  and  furthermore  that  another 
loss  of  head  occurs  in  the  wheel-channels,  between  entrance  A 

cn2 
and  exit  N,  of  an  amount  Cn?r~  (i-e.,  proportional  to  the  square 

. 
of  the  relative  velocity  at  exit)  to  be  placed  in  eq.  (4)  of  §  65. 

These  two  losses  of  head  are  the  h'  and  h",  respectively,  of 
§§  40,  41,  and  42;  £o  and  £n  are  abstract  numbers  (coefficients 
of  resistance;  see  p.  704,  M.  of  E.).  Adopting,  as  -before,  the 
relation  that  for  best  effect  vn  should  be  placed  equal  to  cn,  and 
combining  the  forms  now  assumed  by  eqs.  (3)  and  (4)  with 
the  other  equations  of  §  65  (which  remain  unchanged  in  form), 
we  finally  obtain 


tan  a         \n  sm 
as  the  value  of  vn  for  best  effect;    that  is, 


(21) 


sm 


tanc      .  /          Co  r.*     .sin,?          Cn  tan  q\ 

^^'^smacosa+  2  an*/    (22) 


According  to  Weisbach,  a  value  of  0.05  fo  0.10  may  be  taken 
for  each  of  the  coefficients  £0  and  £n.  If  the  larger  value,  0.10, 
be  taken  and  substituted  in  eq.  (22),  with  ordinary  values  of 
the  ratio  rn:r\,  and  the  angles  a  and  d,  there  results 


,92(>A^Y 

\  X    sin  d    /' 


+*°*\fiErf  • 

which  explains  the  8  per  cent,  reduction,  as  an  allowance  for 
friction,   mentioned  in  the  foregoing  paragraphs. 


106  HYDRAULIC   MOTORS.  §  72. 

The  revolving  wheel  encounters  friction  from  the  adjoining 
tail-water  and  also  at  its  own  axle.  These  various  frictions- 
and  the  fact  of  leakage  of  water  through  the  space  betweea 
the  edges  of  the  wheel-crowns  and  fixed  guides  render  any  refined 
analysis  out  of  the  question.  Only  approximate  results  caa 
be  reached,  short  of  actual  test. 

72.  Efficiency  of  the  Fourneyron  Turbine.  Friction  Con- 
sidered. —  If  we  deduct  the  losses  of  head  just  mentioned  from 
the  whole  head  h  (Fig.  45),  and  also  the  velocity  head  due  to 
the  absolute  velocity  wn  of  the  water  at  exit,  we  have,  for 
the  net  power  (ft.-lbs.  per  sec.), 

2      -   -  -   (24) 


and  therefore,  for  the  efficiency, 
Rfvf     r       w.n2 


For  example,  if  we  substitute  in  this  equation  the  values 

occurring  in  the  last  numerical  problem  (§70),  viz.,  h  =  QO  ft.; 
^ 

Wi=45.4,   wn  =  2vn  sin  —  =  12.5,   and  cn=vn=48,   ft.    per  sec.; 

2i 

with  0.10  for  both  £o  and  £n;  we  obtain 

60-2.5-3.2-3.6     50.7 
9-  60  =  6Q  ==0-84> 

or  84  per  cent.  But  the  pawer  lost  in  axle  friction  (R"vff)  and 
that  spent  on  the  resistance  of  tail-water  on  the  outside  sur- 
faces of  the  crowns,  etc.,  would  probably  reduce  this  to  some 
78  or  even  75  per  cent.  (See  §§99,  etc.,  as  to  actual  tests 
of  turbines.) 

73.  Note.  —  Evidently  the  bracket  in  eq.  (24)  represents 
the  work  done  per  sec.  for  each  unit  of  weight  of  water  used; 
thus,  in  the  numerical  instance  above,  from  each  pound  of 
water  are  derived  50.7  ft.-lbs.  of  work  per  sec.,  out  of  the  total 
theoretical  60  ft.-lbs.  for  each  pound  of  water.  Of  the  total 
loss  (9.3),  2.5  ft.-lbs.  per  sec.  is  due  to  residual  kinetic  energy 


§  74.  FOURNEYRON   TURBINE.  107 

at  exit,  3.2  to  fluid  friction  in  the  penstock  or  wheel-pit,  and 
3.6  to  fluid  friction  in  the  wheel-channels.  . 

74.  Fourneyron  Turbines  at  Niagara  Falls,  N.  Y. — During  the 
years  1894  to  1903  the  Niagara  Falls  Power  Co.  constructed  a 
water-power  " installation"  about  a  mile  above  the  falls  at 
Niagara  Falls,  N.  Y.,  involving  two  power-houses  containing 
twenty-one  turbines,  and  a  tunnel  (as  a  tail-race)  some  6700 
feet  in  length  and  490  sq.  ft.  in  sectional  area,  on  a  grade  of 
7  ft.  per  thousand,  and  at  a  depth  at  its  upper  end  of  some 
146  ft.  below  the  level  of  the  upper  river.  The  tunnel  is  of  a 
horseshoe  form  in  section,  is  lined  with  hard  brick,  and  empties 
at  the  base  of  the  cliff  a  short  distance  below  the  American 
Fall.  The  velocity  of  the  water  in  it  is  sometimes  as  great 
as  25  ft.  per  second. 

In  "'Power  House  No.  1"  each  of  ten  vertical  shafts  carries 
two  Fourneyron  turbines,  each  such  (double)  wheel  or  "unit" 
furnishing  5000  H.P.  and  working  under  a  mean  head  of  136  ft., 
at  250  revolutions  per  minute,  and  using  about  440  cub.  ft. 
of  water  per  second.  The  wheel-pit  under  the  power  house 
is  an  immense  slot  excavated  in  the  rock,  the  lower  part  dis- 
charging the  water  after  its  passage  through  the  turbines 
into  the  upper  end  of  the  tunnel.  Each  of  these  double  wheels 
has  a  separate  penstock  into  which  water  is  admitted  from  a 
wide  canal,  leading  out  of  the  upper  river.  These  turbines  * 
were  built  and  installed  by  the  I.  P.  Morris  Co.,  of  Philadelphia; 
from  designs  by  Faesch  and  Piccard  of  Geneva,  Switzerland. 

Fig.  52  gives  a  side  view,  or  elevation,  of  one  of  these  pen- 
stocks with  its  corresponding  wheel-casing,  shaft,  etc.  The 
steel  penstock,  P,  is  7.5  ft.  in  diameter,  conducting  water 
under  pressure  to  the  wheel-casing,  e.  At  the  upper  and 
lower  extremities  of  this  casing  revolve  the  two  wheels,  the 
discharge  from  which  issues  at  a  from  the  upper,  and  at  T 
from  the  lower,  turbine.  T  shows  also  the  level  of  the  tail- 
water  at  the  bottom  of  the  wheel-pit.  The  height  of  its  sur- 
face is  variable,  depending  on  the  number  of  turbines  in  action 
at  any  time. '  Although  each  turbine  works  in  a  position  above 
the  tail-water,  discharging  into  the  atmosphere,  its  design  is 

*  Replaced  in  1913  by  Francis  turbines  with  draft  tubes. 


FIG.  53. 


•108 


§  74.  FOURNEYRON    TURBINES   AT    NIAGARA.  109 

such  that  all  the  channelways  are  completely  filled  with  water, 
so  that  it  operates  as  a  "reaction  turbine"  and  not  as  an  impulse, 
or  "  tangential/'  wheel. 

S  is  the  shaft,  which  for  strength  and  lightness  combined 
is  mainly  hollow,  consisting  of  segments  of  steel  tubing  38  inches 
in  diameter,  connected  to  each  other  at  intervals  .by  short 
solid  portions,  11  in.  in  diameter,  running  in  bearings.  These 
bearings,  however,  provide  only  lateral  support.  On  the  upper 
end  of  the  shaft  is  fixed  the  revolving  part,  G,  of  an  electric 
generator,  which  has  sufficient  mass,  with  that  of  the  two  tur- 
bines themselves,  to  serve  as  a  fly-wheel. 

In  Fig.  53  is  given  a  section,  on  a  larger  scale,  of  the  lower- 
end  of  the  penstock  and  of  the  wheel-casing  and  turbines. 
Although  the  velocity  of  the  water  in  the  penstock  is  about 
10  ft.  per  second,  the  fluid  pressure  in  the  casing  e  differs  but 
slightly  from  the  hydrostatic  pressure  due  to  the  whole  head 
of  136  ft.  The  two  turbines,  and  their  supporting  shells  extend- 
ing out  from  the  shaft,  are  indicated  by  solid  black  shading 
(better  shown  in  a  subsequent  figure).  Rigidly  attached  to 
the  shaft  S  is  a  disc  M,  the  space  underneath  which  is  in  com- 
munication with  the  water  of  the  penstock,  while  the  upper 
face  is  open  to  the  atmosphere.  The  lifting  effort  thus  exerted 
on  the  shaft  serves  to  sustain  the  greater  part  of  the  weight 
of  the  shaft,  turbines,  and  generator.  In  other  words,  the 
friction  of  a  solid  disc  on  a  liquid  is  substituted  for  that  of  a 
journal,  or  pivot,  in  a  solid  bearing;  a  gain  both  in  convenience 
and  power.  The  excess  (or  deficiency)  of  this  hydrostatic 
pressure  is  taken  up  by  a  special  thrust-bearing  at  the  upper 
end  of  the  shaft. 

The  lower  turbine  of  one  of  these  double  wheels  (or  "units  ") 
is  shown  in  vertical  section  in  Fig.  54,  where  the  solid  black 
shading  indicates  the  revolving  part,  or  turbine  ("'runner") 
itself.  Between  the  crown-plates  E  and  D  are  placed  two 
horizontal  partitions,  thus  practically  dividing  the  turbine 
into  three  separate  turbines  (see  Fig.  46  in  this  connection). 
Corresponding  partitions  are  also  placed  at  G  between  the 
guides.  The  extreme  outside  diameter  of  the  turbine  is  6  ft* 


110  HYDRAULIC   MOTORS.  §  74. 

2  in.;  and  the  inner  diameter  of  the  crowns  is  5  ft.  3  in.  The 
vertical  distance,  e,  between  crowns  is  about  12  inches.  A 
portion  of  the  turbine  and  guides  is  shown  in  horizontal  sec- 
tion in  Fig.  55,  where  it  is  seen  that  the  middle  portions  of 
the  wheel-vanes  are  thickened,  and  in  such  a  way  as  to  secure 
more  gradual  changes  of  cross-section  in  the  wheel-passages 
than  would  otherwise  be  the  case. 

The  regulating-gate  is  a  vertical  cylinder  placed  outside 
of  the  turbine.  It  is  shown  in  horizontal  section  in  Fig.  55; 
and  in  vertical  section,  at  C,  in  Fig.  54,  in  which  latter  figure 
the  gate  is  entirely  closed.  A  downward  motion  of  the  rods 
R,  R,  is  required  to  open  it.  The  corresponding  gate  of  the 
companion  turbine  at  the  upper  part  of  the  casing  (at  a  in 
Fig.  53)  is  moved  simultaneously  by  the  same  rods.  In  this 
way  one  or  more  of  the  spaces  between  the  horizontal  parti- 
tions of  each  turbine  is  opened  for  the  action  of  the  water. 
Though  this  method  of  regulation  is  usually  accompanied  by 
a  low  efficiency  at  "  part  gate/'  the  effect  is  here  much  improved 
by  the  presence  of  the  horizontal  partitions.  The  great  hydro- 
static pressure  on  the  stationary  disc  m  (Fig.  54)  forming  the 
floor  of  the  wheel-casing  is  sustained  by  the  rods  K,  K,  (see 
also  Fig.  55,)  whose  upper  ends  are  fastened  to  the  sides  of 
the  casing.  Each  turbine  contains  32  vanes  (or  "buckets")^ 
while  the  number  of  guides  is  36.  The  angles  a,  /?,  and  d  in 
these  wheels  have  values  of  about  20°,  110°,  and  13°,  respect- 
ively. 

Tests  of  one  of  these  double  turbines  have  shown  an  effi- 
ciency as  high  as  82  per  cent.,  the  useful  power  being  measured 
electrically;  and  the  consumption  of  water  determined  by 
current-meters  held  at  the  entrance  of  the  penstock. 

All  of  the  ten  (double)  turbines  in  Power  House  No.  1, 
each  of  5000  H.P.,  are  situated  in  a  common  wheel-pit  and 
deliver  their  water  into  a  common  tail-race  which  empties 
into  the  upper  end  of  the  great  tunnel.  Each  is  regulated 
to  a  fairly  constant  speed  by  a  governor  of  special  design, 
any  slight  change  of  speed  affecting  the  angular  position  of 
the  centrifugal  "  fly-balls."  With  any  increase  of  speed  from 


GATE 


WHEEL 


FIG.  55. 


in 


112 


HYDRAULIC    MOTORS. 


§75.- 


the  normal  the  gate  mechanism  is  thrown  into  gear  with  the 
turbine  itself  and  the  gate  is  partially  closed  until  the  speed 
returns  to  its  normal  value;  and  vice  versa.  In  this  way  the 
speed  does  not  vary  more  than  3  or  4  per  cent,  from  the  normal,, 
even  when  as  much  as  25  per  cent,  of  the  "load"  (R');  or 
resistance,  is  suddenly  removed.  (In  Power  House  No.  2,. 
of  more  recent  construction,  the  turbines  are  of  another  type;, 
see  §  78.) 

The  turbines  just  described  are  made  chiefly  of  cast  iron,, 
with  some  smaller  parts  of  steel. 

75.  The  Fall  River  Turbine. — The  Fourneyron  turbine,, 
made  at  Fall  River,  Mass.,  by  Kilburn,  Lincoln,  and  Co.,  is- 
shown  in  Figs.  17,  18,  and  19  (opp.  p.  42).  The  nest  of  guides 

fits  within  the  inner 
hollow  of  the  wheel,  or 
"  runner,"  while  the 
cylindrical  gate  is  mov- 
able vertically  between. 
Fig.  56  gives  a  view  of 
the  exterior  of  wheel- 
case,  etc.  The  pen- 
stock is  attached  at  P. 
The  turning  of  the 
small  shaft  H,  by 
means  of  intervening 
screw-gearing,  causes, 
motion  of  the  four  ver- 
tical rods  to  which  the 
gate  G  is  attached.  At 
T  is  seen  the  turbine 
itself.  By  bevel-gear- 
ing the  turbine  shaft- 
communicates  motion, 
at  E,  to  the  horizontal 
shaft  S,  for  the  driving 
FIG.  56.  of  machinery,  etc.  One 

of  these  wheels  was  tested  in  1870  by  Mr.  Clemens  Herschel, 


§  76.  CLASSIFICATION    OF    TURBINES.  113 

and  gave  an  efficiency  of  practically  80  per  cent,  under  a  head 
of  h  =  19.6  ft.;  developing  130.3  H.P.  at  its  "best  speed"  of 
92.5  revs,  per  min.,  and  using  Q  =  58.Q  cub.  ft.  of  water  per 
sec.  The  diameter  of  the  turbine  was  5  ft.  8  in.,  and  height 
of  wheel-passages  e=6.4  in.  These  wheels  are  made  with 
either  iron  or  bronze  buckets. 

76.  Classification  of  Turbines. — A  general  definition  of  a 
turbine  may  be  thus  stated,  viz. :  A  water  motor  consisting  of  a 
number  of  short  curved  pipes  set  in  a  ring  attached  rigidly  to  a 
shaft  upon  which  it  revolves,  and  receiving  water  at  all  parts  of 
its  circumference  from  the  mouths  of  other  and  fixed  pipes 
or  passageways;  the  cross-section  of  all  of  these  curved  pipes 
being  completely  filled  with  water  during  steady  operation. 
Tho  principal  types  of  turbines  are  as  follows: 

I.  Radial,    Outward-flow,     Turbines;     in    the    working    of 
which  the  general  course  of  the  water  lies  in  a  plane  at  right 
angles  to  the  axis  or  shaft  and  is  directed  outward,  away  from 
the  axis  of  rotation.     (The  Fourneyron  turbine  just  treated 
is  of  this  type).     In  this  case  the  guide  blades  serving  to  form 
the    fixed    passageways   are   placed   within    the    turbine    and 
deliver  water  to  the  turbine    channels  at  the  inner  edge  of 
the  turbine-ring. 

II.  Radial,    Inward-flow,    Turbines;    in    which    the    fixed 
guide-passages  are  situated  on  the  outside  of  the  turbine-ring, 
the  general  course  (absolute  path)  of  the  water  in  the  turbine 
channels  lying  in  a  plane  perpendicular  to  the  axis  but  directed 
radially  inward.     These  are  called  Francis,  or  "  center- vent/' 
wheels. 

III.  Axial  Flow,  or  Parallel  Flow,  Turbines;   in  which   the 
absolute  path  of  a  particle  of  water  lies  substantially  in  the 
surface  of  a  cylinder  whose  axis  coincides  with  that  of  the 
turbine;  that  is,  all  points  of  this  path  are  practically  equidis- 
tant from  the  axis  of  rotation.     (The  "  Jonval "  Turbine.) 

IV.  Mixed  Type,  or  Mixed  Flow. — In  case  the  water  enters 
the  turbine  channels  from  the  outside,  having  at  first  a  radial 
and  inward  direction  of  motion,  and  is  later  so  diverted  as  to 
leave  those  channels  in  a  direction  parallel  to  the  axis,  the 


HORIZONTAL 


(rcRUNNER") 
-WHEEL 


FIG.  57. 


»S  77.  CLASSIFICATION    OF  TURBINES.  115 

turbine  is  said  to  be  one  of  Mixed  Type.  Most  American 
turbines  belong  to  this  type  of  wheel.  ("Inward  and  down- 
ward discharge.") 

77.  Radial,  Inward-flow,  Turbine.  (The  Francis  Wheel.) — 
A  simple  arrangement  of  this  type  of  turbine  is  shown  in  Fig.  57. 
in  the  upper  part  of  which  is  a  vertical  section  of  the  wheel, 
•shaft,  casing,  etc.;  while  below  is  a  horizontal  section  taken 
through  a  point  half-way  between  the  crowns  of  turbine.  The 
section  of  the  turbine  crowns,  shaft,  and  supporting  shell,  k, 
are  shaded  in  solid  black.  The  fixed  guides  are  placed  in 
the  space  G,  on  tho  outside  of  the  turbine,  while  the  curved 
vanes  of  the  turbine  are  situated  between,  and  unite,  the  two 
•crowns  a  and  e.  After  passing  through  the  turbine-ring  the  water 
finds  its  way  through  the  vertical  tube  eSe,  and  finally  joins 
the  tail-water  at  T.  At  the  upper  end  of  the  shaft,  which  pro- 
trudes through  the  upper  floor,  D,  of  the  water-tight  wheel- 
casing  M,  is  keyed  a  pulley,  F,  at  whose  circumference  a  resist- 
ance, Rf  Ibs.,  is  overcome  at  a  velocity  v'  ft.  per  sec.  By  a 
downward  movement  of  the  ring  m  the  sectional  areas  between 
the  guides  may  be  reduced;  when  less  water  is  to  be  used. 
The  horizontal  plate  u,  supported  by  rods  from  above,  serves 
to  protect  the  revolving  plate  k  of  the  wheel  from  the  high 
pressure  in  the  space  M,  where  the  water  is  slowly  travelling 
toward  the  guide-openings  at  G.  The  surface  of  the  head- 
water is  not  shown,  being  at  an  elevation  above  the  upper 
floor  D  (of  the  casing),  which  is  therefore  subjected  to  con- 
siderable hydrostatic  pressure  from  the  water  in  space  M 
underneath. 

The  theory  of  the  inward-flow  turbine  does  not  differ  essen- 
tially from  that  of  the  outward-flow  type  already  given  (see 
§  89,  etc.,  where  a  general  theory  for  all  turbines  will  be  given). 
It  will  be  sufficient  for  the  moment  (see  Fig.  58)  to  note  the 
parallelograms  of  velocities  at  entrance  (point  1),  and  at 
•exit  (point  N)  in  the  inner  circumference  of  wheel.  The 
same  notation  is  used  as  in  the  case  of  the  outward-flow 
turbine;  that  is,  the  subscript  1  refers  to  the  point  of  entrance 
and  N  to  that  of  exit.  1  ...  N  is  the  absolute  path  of  the 


116 


HYDRAULIC    MOTORS. 


§78, 


FIG.  58. 


water  in  passing  through  the  wheel-ring,  and  for  best  effect, 
after  the  "best"  value   of  the  exit  wheel-rim  velocity  vn  has 

been  determined,  and 
the  corresponding  value 
of  the  absolute  velocity 
Wi  at  point  1  of  leaving: 

A  O" 

the  guides,  the  tangent 
to  wheel-vane  at  1  must 
be  so  placed  as  to  coin- 
cide with  the  position 
of  the  relative  velocity 
ci  as  determined  by  the 
values  of  vi  and  w\ 
already  found.  There 
will  then  be  no  sudden 
change  of  direction  in  the  absolute  path  of  the  water  at  the 
point  1,  at  the  entrance  of  the  wheel-channels,  and  hence 
no  "  shock"  and  accompanying  loss  of  energy. 

78.  Francis  Turbines  at  Niagara  Falls.*— In  their  "Power 
House  No.  2"  the  Niagara  Falls  Power  Co.  has  recently  installed 
eleven  turbines  of  about  5500  H.P.  each,  substantially  of  the 
Francis  type.  Fig.  59  gives  an  end  view  of  the  wheel-pit  show- 
ing the  penstock,  shaft,  etc.,  of  one  of  the  turbines.  S  is  the 
turbine  shaft,  chiefly  tubular  (3.28  ft.  in  diameter;  of  metal 
|  in.  thick),  with  occasional  solid  portions  for  lateral  support,, 
in  bearings.  Behind  the  shaft  is  seen  the  penstock,  P,  P, 
into  which  the  water  enters  at  H  from  the  upper  river.  The 
penstock  is  made  of  steel  plates  J  in.  in  thickness  and  is  7  ft. 
6  in.  in  diameter;  and  conducts  the  water  to  the  turbine  in 
the  wheel-casing,  E.  After  leaving  the  wheel,  the  water  enters 
the  upper  ends  of  two  "draft-tubes"  (or  "suction-tubes,"  as 
they  are  often  called),  from  which  it  is  finally  discharged  into 
the  tail- water  at  T.  These  "'draft-tubes"  discharge  under 
water  that  the  air  may  not  enter  and  thus  prevent  their  flowing 
full.  They  act  like  water-barometers,  except  that  the  water 

*  See  the  Engineering  Record,  Nov.  1901,  p.  500;    also  Nov.  and  Dec. 
1903,  pp.  616,  652,  691,  and  763. 


SHAFT 


FIG.  59. 


118  HYDRAULIC    MOTORS.  §  78. 

is  in  motion,  the  internal  fluid  pressure  being  less  than  one- 
atmosphere  at  points  of  higher  elevation  than  the  surface  of 
the  tail-water.  Their  upper  extremities  are  not  more  than, 
about  20  ft.  above  the  surface  of  the  tail-water,  so  that  the 
water  continues  to  fill  the  tubes  after  the  air  has  once  been. 
swept  out  of  the  tubes  by  the  current.  The  draft-tubes  are 
placed  within  the  walls  of  the  wheel-pit  in  order  that  they 
may  not  obstruct  the  flow  of  water  from  the  other  turbines  on 
its  way  to  the  junction  (at  one  end  of  the  wheel-pit)  with  the 
great  tunnel  which  serves  as  a  tail-race  for  both  power  houses. 
By  this  arrangement,  also,  the  whole  head  of  some  ]46  ft.. 
from  the  head-water  to  the  surface  of  the  tail-water  is  made 
effective. 

The  interior  of  the  wheel-case  and  draft-tubes,  etc.,  is  shown 
by  the  vertical  section  of  Fig.  60  (largely  diagrammatic). 
The  water  from  the  penstock  fills  the  annular  chamber  A 
under  nearly  hydrostatic  pressure,  passes  through  the  guide- 
passages  at  G,  and  enters  the  wheel-channels  at  c  under  reduced 
pressure  and  with  high  velocity.  The  revolving  turbine,  shaft, 
and  attachments  are  shown  in  solid  black  shading  (except 
the  portion,  S,  of  the  first  tubular  part  of  shaft).  There  are 
25  guide-blades  in  the  ring  G  surrounding  the  wheel;  the  blades* 
and  ring  being  of  bronze,  cast  in  one  solid  piece.  The  wheel 
itself,  also  of  bronze  and  cast  in  one  piece,  contains  21  vanes 
or  buckets  in  the  space  extending  from  c  about  half-way  to  D, 
is  5.25  ft.  in  diameter,  and  is  operated  at  a  speed  of  250  revs, 
per  min.  The  water  leaving  the  turbine-channels  enters  the 
space  D  with  both  low  (absolute)  velocity  and  low  pressure, 
the  pressure  being  practically  that  corresponding  *  to  the 
height  of  D  above  the  tail-water  surface  (which  is,  however,, 
variable  in  position).  At  the  lower  extremity  of  the  shaft,, 
while  lateral  support  is  provided  by  the  bearing  or  step  at  Rr 
a  great  lifting  force  is  furnished  by  the  admission  of  water 
under  the  full  penstock  pressure  to  the  space  U,  U,  on  the 
under  side  of  the  conical  shell,  or  piston  V,  V,  or  "  balancing 

*  E.g.,  if  that  height  were  22  ft.,  the  pressure  would  be  about  5  Ibs.  per 
sq.  in.,  only. 


§   79.  THE   FRANCIS  TURBINE.  110 

disc"  keyed  upon  the  shaft  and  revolving  with  it.     The  pres- 
sure on  the  upper  surface  of  this  piston  is  small,  of  course, 
being  that  of  the  water  in  the  upper  end  of  the  draft-tube. 
In  this  way  the  larger  part  of  the  weight  of  the  wheel,  shaft/ 
and  armature  of  the  electric  generator  is  supported  by  fluid 
pressure.     The  diameter  of  this  piston  or  "balancing  disc"  is-. 
4.9  ft.     The   turbine   was  cast,   in  one   piece,   of  manganese- 
bronze  and  weighs  4000  Ibs.  nearly.     The  weight  of  the  whole 
revolving  mass,  including  that  of  the  armature  of  the  generator,, 
is  71  tons,  to  sustain  which  the  pressure  underneath  the  "balanc- 
ing disc  "  provides  an  upward  force  of  some  66  tons,  leaving 
about  5  tons  to  be   sustained  by  a  "suspension  bearing"  at 
the  upper  end  of  the  shaft. 

The  gate  of  the  turbine  is  a  cast-steel  ring  or  cylinder  mov- 
ing vertically  in  the  narrow  space  c  (Fig.  60)  between  guides 
and  wheel,  and  operated  by  rods  through  the  space  g.  It  is 
not  shown  in  the  figure.  These  eleven  turbines  were  installed 
by  the  I.  P.  Morris  Company  of  Philadelphia  after  designs  of 
Escher,  Wyss  and  Co.  of  Zurich,  Switzerland.  Other  large 
•Francis  turbines  (10,000  H.P.  each)  are  in  use  by  a  branch 
company  on  the  Canadian  side  at  Niagara  Falls. 

79.  Other  Large  Francis  Turbines. — The  Shawinigan  10,500- 
H.P.  turbine  was  designed  and  constructed  in  1904  by  the 
I.  P.  Morris  Company,  and  installed  at  Shawinigan  Falls, 
in  the  Province  of  Quebec,  Canada.  It  is  also  of  the  Francis 
inward-flow  type.  A  view  of  the  wheel-case  and  the  upper 
segments  of  the  draft-tubes  is  given  in  the  frontispiece  of 
this  book.  The  penstock  joins  the  wheel-case  at  the  lower 
left-hand  corner.  The  wheel-case  is  of  spiral  (or  "volute") 
form,  the  space  for  the  water  being  progressively  narrowed 
in  the  circuit  around  the  ring  containing  the  guides.  As 
evident  from  the  figure  the  turbine  revolves  in  a  vertical  plane, 
its  shaft  being  horizontal.  The  water  leaving  the  turbine 
toward  the  center  passes  into  the  two  draft-tubes,  the  upper 
curved  segment  of  one  of  which  is  seen  in  the  figure.  The 
hydraulic  cylinders  at  the  top  furnish  power  for  moving  the 
regulating  apparatus.  In  this  design  the  guide-blades  are 


120  HYDRAULIC    MOTORS.  §  80. 

movable  about  their  inner  ends,  as  in  the  Thomson  Vortex 
Wheel  (see  §  80),  and  by  their  change  of  position  the  area 
of  cross-section  of  the  guide-channels  is  varied  and  thereby 
the  quantity  of  water  per  second  controlled:  The  movable 
guide-vanes  are  operated  by  circular  rings,  and  these  rings  by 
the  pistons  of  the  hydraulic  cylinders.  The  turbine  or  "run- 
ner" is  cast,  in  one  piece,  of  an  alloy  of  about  88  parts  copper, 
10  parts  tin,  and  2  parts  zinc,  and  has  an  external  diameter 
of  7  ft.  It  operates  under  a  head  of  135  ft.  The  I.  P.  Morris 
Co.  is  also  building  (June  1905)  four  turbines,  each  of  13,000 
H.P.,  for  a  Canadian  company  at  Niagara  Falls.  Each  of 
these  consists  of  two  wheels  of  the  Francis  type  mounted  on 
one  shaft  and  discharging  into  one  central  "draft-chest." 
Each  "  runner  "  is  fitted  with  solid  cast  guides,  with  cast-steel 
cylinder-gates  and  bronze  wheels,  the  inside  diameter  of  the 
C}dinder-gate  being  about  5  ft.  5  in.  The  diameter  of  the 
supply-pipe  or  penstock  is  10  ft.  6  in.;  and  that  of  the  draft- 
tube  9  ft. 

The  two  wheels  above  described  are  probably  the  largest 
turbine  "units "  that  have  been  built,  up  to  the  present  date 
(Sept.  1905). 

In  Fig.  60a  is  shown  a  3000-H.P.  turbine  intended  for  a 
power  station  at  Glommen,  Norway;  designed  and  constructed 
by  Escher,  Wyss  and  .Co.  of  Zurich,  Switzerland.  The  runner 
itself  is  on  the  right.  This  engraving  is  from  a  pamphlet 
published  by  the  Allis-Chalmers  Company,  American  agents 
for  the  above-mentioned  Swiss  firm  and  manufacturers  of  its 
designs.  (See  also  Fig.  60&,  opp.  p.  124.) 

80.  The  Thomson  Vortex  Wheel  is  also  of  the  radial  inward- 
flow  type,  and  was  invented  by  Prof.  James  Thomson.  It 
is  remarkable  for  its  excellent  device  for  regulating  the  flow 
of  water  and  for  the  fact  that  the  outer  radius  is  made  from 
two  to  four  times  as  great  as  that  of  the  inner,  or  discharge, 
circumference.  Fig.  61  shows  a  view  of  one  of  these  wheels, 
one-half  of  the  upper  plate  of  the  wheel-case  being  removed. 
From  the  space  within  the  outer  casing  the  water  finds  its  way 
into  four  gradually  contracting  passages,  A,  A,  etc.,  leading 


§81. 


PARALLEL-FLOW   TURBINE. 


121 


to  the  wheel-entrances;  i.e.  there  are  only  four  guide-blades, 
like  RG.  Each  of  these  guide-blades  is  pivoted  at  G,  very  near 
the  extremity,  so  that  when  the  blades  are  turned  on  these 
pivots,  the  water  way  may  be  diminished  in  sectional  area; 
without,  however,  sensibly  altering  the  general  form  of  the 
stream,  thus  avoiding  any  sudden  enlargement  of  its  section 
at  entrance  of  wheel  with  the  consequent  loss  of  energy.  The 
water  leaves  the  wheel  at  E. 

This  wheel  is  very  efficient,  and  is  to  a  certain  extent  self- 
regulating  in  the  matter  of  speed;  for  if,  through  lightening 
•of  load,  the  speed  becomes  augmented,  the  "  centrifugal  action77 
of  the  water  between  the  wheel-vanes  tends  to  "oppose  the 


o, 

Q. 


FIG.  61. 

entrance  of  water  from  the  supply-chamber";  and  vice  versa 
(from  a  report  of  Prof.  Rankine  on  this  wheel). 

81.  The  Parallel-flow  (or  Axial)  Turbine,  usually  called 
the  Jonval  wheel.  (It  is  sometimes  named,  however,  after 
Fontaine,  Henschel,  and  Koecklin,  according  to  slight  dif- 
ferences in  minor  details.) — In  this  turbine,  as  in  the  two 
preceding  types,  fixed  guides  deliver  the  water  without  impact 
into  the  wheel-passages,  whose  vanes  are  curved  in  such  a 
manner  (in  connection  with  a  proper  speed  of  wheel)  that 
the  final  absolute  velocit}^  wn  is  as  small  as  possible;  but  the 
water  passes  through  the  wheel  in  cylindrical  surfaces  sub- 


122  HYDRAULIC    MOTORS.  §  SI- 

stantially  parallel  to  the  axle  or  shaft.  Hence  this  type  of 
wheel  resembles  somewhat  a  screw-propeller  of  numerous 
blades;  bounded  by  two  concentric  cylindrical  shells. 

In  Fig.  62  is  shown  a  vertical  section  of  the  shaft,  pen- 
stock, and  discharge-tube  (or  "draft-tube,"  if  turbine  is  above 
tail- water)  of  a  parallel-flow  turbine.  The  sides  of  the  dis- 
charge-tube T,  T,  are  in  this  case  rigid  prolongations  of  those 
of  the  (vertical)  penstock  or  tube  P,  P.  S,  S,  is  the  shaft, 
to  which  the  turbine  W  is  rigidly  attached.  G  is  the  side 
view  of  the  guide-box  or  fixed  ring  containing  the  stationary 
guide-channels  formed  by  the  guide-blades  and  two  concentric 
cylindrical  walls,  BE  and  AC.  AB  ( =e0)  is  the  radial  width  of 
this  ring.  (In  Fig.  64  the  running  part,  wheel  and  shaft,  is 
shown  in  wide  black  lines.)  The  mouths  of  the  guide-channels 
are  open  all  around  the  ring  AB  and  deliver  the  water  into 
the  channels  of  the  turbine,  W,  below.  The  turbine  is  itself 
a  ring  of  channels  receiving  water  above  and  discharging  it 
below.  In  this  figure  the  width,  en,  of  the  turbine  ring  at 
the  point  of  exit  is  equal  to  that,  eQ,  at  the  point  of  entrance. 
But  frequently  en  is  made  larger  than  e0,  thus  producing  a 
"bell-mouthed,"  or  flaring,  shape  for  the  axial  section  of  the 
wheel-passages. 

Let  now  a  cylindrical  cutting  surface,  aa,  having  its  axis 
in  that  of -the  shaft,  be  imagined  to  be  passed  through  points 
half-way  out,  radially,  between  the  vertical  walls  of  the  guide- 
ring;  its  radius  is  the  ">"  of  Fig.  62. 

The  intersections  made  by  this  cutting  surface  with  a  few 
of  the  guide-blades  and  turbine-vanes  (these  sections  being 
drawn  in  solid  black  lines)  are  shown,  developed,  in  the  middle 
of  Fig.  62.  The  absolute  path  of  a  particle  of  water  entering 
the  guide-channel  at  H  is  H  ...  1,  through  a  guide-channel; 
and  1  .  .  .  N,  through  the  moving  wheel;  whose  velocity  is 
supposed  to  be  such,  together  with  a  proper  value  for  the 
angle  /?,  that  there  is  no  impact,  or  "shock,"  at  1,  the  entrance 
to  a  turbine  channel;  that  is,  that  the  whole  absolute  path 
H  ...  1  ...  N  (dotted)  is  a  smooth  curve,  without  sudden, 
turn  or  elbow  at  point  1. 


FIG.  62. 


FIG.  63. 


123 


124  HYDRAULIC    MOTORS.  §  82. 

The  points  1  and  N  are  half-way  out  along  the  radial  dimen- 
sion of  the  turbine  ring,  being  at  a  radial  distance  =r  from 
the  axis.  The  linear  velocities  of  these  two  points  are  of 
course  equal;  that  is,  vn  =  vi.  The  notation  used  in  Fig.  62 
for  the  parallelograms  of  velocities  at  entrance  and  exit  is 
the  same  as  in  previous  figures;  Wi  and  wn  being  the  absolute 
velocities;  c\  and  cn,  the  relative;  while  v\  and  vn  are  the 
turbine  (linear)  velocities  at  the  points  in  question.  Each 
of  these  parallelograms  lies  in  a  plane  (vertical,  here)  tangent 
to  the  cylindrical  cutting  surface  above  mentioned. 

The  curved  plate  or  shell  m .  . .  m  prevents  the  passage  of 
the  water  from  the  penstock  to  the  turbine  except  through 
the  guide-channels.  See  also  Fig.  64,  where  a  pulley,  or  gear- 
wheel, is  shown  keyed  on  the  upper  end  of  the  shaft;  the 
resistance  Rf  acting  at  the  circumference  of  this  wheel  is  over- 
come through  a  distance  v'  each  second  by  the  action  of  the 
couple  formed  by  the  horizontal  components  of  the  water 
pressures  on  the  turbine-vanes.  The  vertical  components 
create  a  downward  thrust  on  the  turbine  supports. 

82.  The  Draft-tube. — Jonval  was  the  first  to  discover  that 
a  turbine,  especially  his  own,  occupying  so  little  space  hori- 
zontally, would  operate  with  practically  the  same  efficiency 
when  placed  above  the  level  of  the  tail-water  and  discharging 
its  water  into  the  upper  end  of  a  "draft-tube"  or  air-tight  tube 
opening  below  the  water  surface  of  tail-water.  So  long  as 
the  internal  fluid  pressure  of  the  water  can  be  kept  greater 
than  zero  the  tube  will  keep  full,  but  for  this  result  to  be  attained 
the  turbine  must  not  be  placed  more  than  about  25  ft.  above 
the  surface  of  the  tail-water. 

Draft-tubes  are  rarely  made  longer  than  10  to  15  ft.,  their 
principal  use  being  to  render  the  turbine  easily  accessible  for 
examination,  repairs,  etc.  Fig.  63  (on  p.  123)  shows  a  wheel- 
casing  receiving  water  from  a  penstock  (not  shown;  entering 
the  casing  from  behind)  and  containing  two  turbines  fixed 
upon  a  common  horizontal  shaft.  Each  of  these  turbines 
discharges  water  into  a  separate  draft-tube.  On  account 
of  the  symmetrical  arrangement  of  the  turbines  the  end  thrusts 


FIG.  6ob.     Escher,  Wyss  &  Co.     Turbines  at  Spiez,  Switzerland. 


FIG.  70.     Victor  High-Pressure  Runner. 


§  82.  THE    DRAFT-TUBE.  125 

along  the  shaft  neutralize  each  other  so  that  only  lateral  friction 
is  occasioned  in  the  shaft-bearings. 

In  the  analysis  of  §  65  (Fourneyron  turbine)  it  is  noticeable 
that  the  results  obtained  are  independent  of  the  depth  hn  of 
the  wheel  below  the  surface  of  the  tail- water.  A  negative 
hn  would  mean  that  the  exit-point  of  the  turbine  is  above  the 
tail-water,  and  in  order  that  the  tube  into  which  it  discharges 
may  flow  full  (after  being  once  cleared  of  air)  it  is  only  neces- 
sary that  its  vertical  length  shall  not  exceed  that  of  the  water- 
barometer  (or,  rather,  something  less;  since  the  water  in  the 
tube  has  a  certain  velocity  and  a  loss  of  head  due  to  skin  fric- 
tion occurs.  (See  §  532  on  the  siphon,  p.  735,  M.  of  E.) 

When  this  air-tight  tube  is  provided,  the  virtual  surface 
of  the  tail- water  is  about  34  ft.  (at  sea  level)  higher  than  the 
actual,  and  a  similar  statement  is  true  for  the  head-water. 
The  " potential  head"  or  " elevation  head"  apparently  lost  by 
the  placing  of  the  turbine  above  the  actual  surface  level  of 
the  tail- water  (within  the  limit  indicated)  is  made  good  by  the 
diminution  of  pressure  (i.e.,  of  ''pressure-head")  at  the  point 
where  the  water  leaves  the  turbine  channels. 

The  only  additional  source  of  loss  of  energy  attending  the 
use  of  the  draft-tube,  as  compared  with  that  occurring  when 
the  turbine  discharges  into  a  large  water-filled  space  below 
the  level  of  the  tail- water,  is  the  loss  of  head  due  to  "'skin 
friction"  in  the  draft-tube  itself,  but  this  ma*y  be  made  quite 
small  if  the  tube  is  sufficiently  short  and  wide  and  the  velocity 
of  the  water  in  it  correspondingly  slow.  If  the  tube  has  the 
same  width  at  the  top  (i.e.,  at  the  exit  of  the  turbine  channels) 
as  elsewhere,  there  is  thereby  produced  a  "'sudden  enlargement" 

A 

of  section  and  a  loss  of  head  whose  value  is  ~L  (from  Borda's- 

2# 

Formula,  p.  721,  M.  of  E.);  the  same  as  if  no  draft-tube  were 
used. 

Draft-tubes  may  be  ehiployed  with  any  class  of  turbiner 
though  the  Jonval  and  Francis  types,  with  their  modifications, 
are  best  adapted  to  its  use,  and  have  even  been  fitted  to 
impulse-wheels  of  the  Pelton  and  Girard  designs.  But  in 


126  HYDRAULIC    MOTORS.  §  83. 

this  latter  instance  the  wheel  revolves  in  rarefied  air  within  a 
strong  casing  forming  the  top  of  the  draft-tube;  the  upper 
surface  of  the  water  in  the  draft-tube  being  maintained 
automatically  just  below  the  lowest  sweep  of  the  moving  buckets. 
An  advantage  secured  in  such  a  case  lies  in  the  diminished  re- 
sistance of  the  air. 

83.  The    Diffuser. — In    previous    paragraphs,     when    the 
statement  has  been  made  that  the  water  carries  away  with  it 

at  exit  a  kinetic  energy  of  — — «r  ft.-lbs.  each  second,  it  was 

i/ 

with  the  understanding  that  the  pressure  at  that  point  was 
that  due  to  a  head  of  34  ft.  (water-barometer  height)  in  case 
the  pressure  around  the  jet  was  that  of  one  atmosphere;  or 
that  due  to  a  depth  hn  of  still  water  (with  atmosphere  on  sur- 
face) between  the  point  of  exit  and  the  surface  of  tail-water. 
But  if  the  current  leaving  the  turbine  channels  does  not 
immediately  enter  a  large  body  of  comparatively  still  water, 
but  is  guided  by  the  rigid  walls  of  a  stationary  and  gradually 
enlarging  passageway,  at  the  entrance  of  which  the  sectional 
area  is  equal  to  that  of  the  current;  then  the  internal  fluid 
pressure  at  the  point  of  exit  from  the  turbine  is  not  that  corre- 
sponding to  the  position  of  this  point  (hydrostatically)  with 
reference  to  the  surface  of  the  tail-water,  but  will  be  less  (pro- 
vided the  tube  conducting  the  water  from  the  turbine-exit  to 
the  main  body  of  the  tail- water  is  of  proper  design).  Such 
an  apparatus  to  provide  a  gradual  enlargement  of  section 
for  the  passage  of  the  water  after  it  leaves  the  turbine  is  called 
a  diffuser  and  was  first  invented  and  used  by  Mr.  Boyden  of 
Boston,  Mass.,  about  1845,  in  connection  with  a  radial  out- 
ward-flow turbine.  Its  use  was  found  to  increase  the  efficiency 
of  the  turbine  some  three  per  cent.,  by  actual  experiment. 
In  the  case  of  this  type  of  wheel  the  diffuser  consisted  of  two 
fixed  conical  zones  flaring  out  opposite  the  outer  edges  of  the 
turbine  crowns,  giving  a  "  bell-mouthed"  or  divergent  profile 
to  the  walls  of  the  passageway  at  that  point  of  the  flow.* 

*  Somewhat  as  shown  between  n  and  m  in  Fig.  24,  p.  50,  but  with  a 
much  more  gradual  increase  of  section. 


§84. 


THE    DRAFT-TUBE   AND    DIFFUSER. 


127 


(Kneass's  book  on  the  steam-injector  gives  an  account 
•of  experiments  on  the  flow  of  water  in  divergent  tubes  (i.e., 
in  tubes  of  gradually  enlarging  longitudinal  profile)  which 
are  interesting  in  this  connection). 

When  a  diffuser  is  provided,  the  "  pressure  energy"  carried 
away  by  the  water  at  exit  from  the  turbine  is  smaller  than 
otherwise;  and  the  gain  in  that  respect  aids  in  offsetting  the 
loss  of  energy  due  to  the  water  leaving  the  wheel  with  an  absolute 
velocity  wn.  In  brief,  any  prevention  or  lessening  of  loss  of 
head,  either  in  penstock,  wheel-channel,  or  draft-tube,  is  a 
distinct  gain  to  the  efficiency  of  the  turbine  and  its  appur- 
tenances. 

84.  Theory  of  the  Draft-tube,  with  Diffuser.  —  Fig.  64 
;shows  a  vertical  section  of  a  Jonval  turbine  revolving  on  a 
vertical  shaft  and  pro- 


vided  with  a  draft- 
tube,  DM,  and  a  dif- 
fuser (stationary),  nKn. 
The  revolving  wheel  and 
shaft  are  shown  in  solid 
black  shading.  It  is 
revolving  uniformly  at 
best  speed  and  the  flow 
of  thewater  is  "  steady"  ; 
the  power  developed 
being  employed  in  over- 
coming the  resistance 
Rf  Ibs.  through  a  dis- 
tance v'  each  second  at 
the  periphery  of  the 
pulley,  or  gear-wheel, 
Iveved  upon  the  upper 
end  of  the  shaft;  i/ 
being  the  linear  velocity 
•of  the  periphery  of  the 
pulley. 


H 


FIG.  64. 


The  absolute  velocity  of  the  water,  which  has  a  value  wn 


128  HYDRAULIC    MOTORS.  §  84, 

at  the  point  n  where  it  leaves  the  turbine  channels,  is  gradually 
reduced  to  a  low  value  w'  in  the  cylindrical  part  of  the  draft- 
tube;  and  the  water  finally  leaves  the  tube  at  n",  beneath  the 
surface  T  of  the  tail-water,  through  a  vertical  cylindrical  open- 
ing with  a  velocity  w"  (which  should  be  small,  the  opening  being 
large)  and  under  a  pressure  (pa  +  h"f)  due  to  the  depth  It'  of 
still  water  below  the  surface  T  of  tail-water.  That  is  to  say, 
at  the  point  where  the  water  leaves  the  whole  apparatus,  flow- 
ing into  the  full  body  of  the  tail-  water,  and  where  it  is  under 
a  pressure  corresponding  (hydrostatically)  to  the  depth  of  this 
point  below  the  surface  of  the  tail-water,  its  absolute  velocity 
is  (by  proper  design)  smaller  than  that  at  the  point  n  of  exit 
from  the  turbine  channels  and  the  kinetic  power  thus  carried 
away  correspondingly  small.  The  gain,  however,  in  this 
respect  would  be  more  than  offset  by  the  loss  of  head  between 
n  and  n"  if  the  change  of  absolute  velocity  from  wn  to  w'  were 
not  brought  about  gradually  by  the  gradual  change  of  sec- 
tional area  of  waterway  between  n  and  n'. 

Let  h  and  In!'  denote  the  elevations  so  indicated  in  Fig.  64r 
d'  and  I'  the  diameter  and  length  of  the  cylindrical  portion 
of  the  draft  -tube  (the  tube  is  not  necessarily  vertical),  and  / 
the  coefficient  of  fluid  friction  in  the  same  (p.  707,  M.  of  E.); 
also  let  F'"  (  =  2xren  of  Fig.  62)  denote  the  sectional  area  of 
the  horizontal  ring  at  n  (to  which  the  direction  of  the  velocity 
wn  is  practically  perpendicular  in  the  regular  running  of  the 
turbine  at  its  best  speed),  and  a"  the  altitude  of  the  cylindrical 
opening  at  n"  .  The  small  loss  of  head  due  to  the  gradual 
enlargement  of  waterway  from  n  to  nf  may  be  represented  by 


77} 


-  * 


-i 

*5T*    while,  as  in  a  previous  paragraph,  Co        anr*  C«~<~  are 


< 


those  occurring  in  the  guide-channels  and  wheel-channels 
respectively  (see  §  71  and  Fig.  62).  If  we  now  deduct  *  the  total 
energy  possessed  by  the  flow  per  second  at  point  n"  from  that 
at  point  H,  we  obtain  for  the  power  spent  in  overcoming  the- 
resistance  R'  each  second 


And  also  deduct  the  lost  power  due  to  the  intervening  losses  of  head. 


§  84.  DRAFT-TUBE    AND    DIFFTJSER.  129 

In  this  connection  we  have,  of  course, 

.....     (2) 

(in  which  d"  is  the  diameter  of  the  horizontal  circle  formed  by 
the  edge  n"). 

In  order  that  the  flow  of  water  may  fill  all  sections  of  the 
draft-tube,  as  supposed  in  the  above  analysis,  it  is  necessary 
that  during  the  steady  flow  the  fluid  pressure,  pn,  at  the  point 
71  of  exit  from  the  turbine  channels  be  greater  than  zero;  in 
other  words,  that  the  algebraic  expression  for  this  pressure 
must  not  lead  to  a  negative  result  when  applied  in  any  numerical 
case.  Since  between  points  n  and  n"  the  steady  flow  of  the 
water  takes  place  through  rigid  and  stationary  pipes,  the 
application  of  Bernoulli's  Theorem  for  such  a  case  is  warranted 
and  leads  readily  to  the  following  relation  (with  n"  as  a  datum 
plane;  b  denoting  the  height  of  the  water-barometer  or  about 
34  ft.): 


k,,+bi 

or 


d    2g       2g  2g 

The  value  of  £  may  be  taken  as  approximately  0.30  (see 
§  107).  It  is  evident  from  eq.  (4)  that  the  value  of  the  pres- 
sure pn  would  be  negative  if  the  elevation  hr  were  numerically 
greater  than  the  quantity  in  the  large  bracket;  that  is,  the 
greatest  permissible  value  of  hf  would  be,  theoretically, 


if  flow  with  full  sections  is  to  be  realized.  But  practically, 
since  wrater-vapor  might  form  in  the  upper  part  of  the  tube 
if  the  pressure  were  too  low,  especially  in  a  warm  climate,  this 
value  of  h'  should  not  be  approached  within  (even)  5  or  6  feet. 
If  the  diffuser  were  absent,  that  is,  if  no  provision  were 
made  to  secure  a  gradual  enlargement  of  waterway  between 


130  HYDRAULIC     MOTORS.  §  85. 


n  and  n',  the  term  ~^~}  or  0.30-^,  would  be  replaced  by 


Wr? 


85.  Turbines  of  the  Mixed-flow  Type,  or  "  Inward  and  Down- 
ward." —  In  turbines  of  this  type  the   wheel-channels,   while 
receiving  the  water  from  guides  on  the  outside,  so  that  the 
course  of  the  water  is  at  first  radial  and  inward,  toward  the 
.shaft,  gradually  curve  in  such  a  way  that  at  exit  the  water 
has  an  absolute  path  nearly  parallel  to  the  axle.     The  axle 
being  usually  vertical,  the  course  of  the  water  may  be  rudely 
described  as  "inward  and  downward,"  and  the  turbine  is  said 
to  be  one  of  "mixed  flow." 

Most  American  turbines  are  of  this  class,  a  typical  vertical 
section  (or,  rather,  a  section  through  the  axis  of  the  shaft) 
...WHEEL  being    shown    diagrammatically    in 

Fig.  65,  where  the  solid  black  shading 
represents  the  revolving  part,  or 
"runner."  The  guides  are  placed 
in  a  ring  surrounding  the  "runner" 
GUIDES  ..  B  .-•'*'  as  m  the  Francis  turbine;  but  the 

FlG.  65.  water  leaves  the  turbine  mainly  in  a 

direction  parallel  to  the  axle  or  shaft. 

86.  American     Turbines,  —  A     prominent     and    successful 
American  turbine  is  made  by  the  Risdon-Alcott  Co.,  of  Mount 
Holly,  N.  J.     Fig.  66  shows  the  "runner,"  or  turbine  itself, 
which  has  a  curved  upper  crown;   the  place  of  a  lower  crown 
being  taken  by   a  vertical   cylindrical  band   (represented   as 
transparent  in  the  engraving).     In  Fig.  67  is  seen  an  outside 
view  of  the  wheel-case,  etc.     The  guide-vanes,  B,  B,  etc.,  are 
fixed  upon  the  ring  R.     S  is  the  short  discharge-tube,  intended 
to  dip  a  few  inches  below  the  surface  of  the  tail-  water.     Ihe 
gate  is  a  vertical  cylinder,  seen  at  (7,  and  in  this  make  of  tur- 
bine is  furnished  with  a  number  of  horizontal  extension  pieces, 
such  as  D,  D,  etc.,  accompanying  the  gate  in  its  vertical  move- 
ment, and   providing,  therefore,  a  movable   "roof"   for  each 
guide-channel.     The  turbine  vanes  or  blades  are  warped  sur- 


FIG.  66. 


FIG.  67 


132 


HYDRAULIC    MOTORS. 


§  86. 


faces,  their  lower  extremities  suggesting  the  form  of  a  spoon, 
or  scoop.  The  turbine,  generally  of  cast  iron,  though  some- 
times of  bronze,  is  cast  in  a  solid  piece.  In  Fig.  67,  V  is  the 
shaft  of  the  turbine,  while  the  smaller  shaft  W  is  intended 
to  operate  the  gate;  whose  motion  up  or  down,  as  may  be 
needed,  is  brought  about  through  the  intervening  gear-wheels 
and  rack  by  the  turning  of  shaft  W.  A  wheel  of  this  design 
made  the  highest  record  at  the  turbine  tests  conducted  at 
the  Centennial  Exposition  held  at  Philadelphia  in  1876;  its 
performance  at  part  gate  being  remarkable  for  that  date. 

The  Risdon-Alcott  Co.  also  manufacture  a  turbine  pro- 
vided with  a  "'register"  gate;  which  consists  of  a  cylindrical 
shell  placed  between  the  guides  and  outer  edge  of  wheel  and 
perforated  with  slots  parallel  to  the  shaft,  somewhat  like  a  grid- 
iron. Fig.  67a  shows  such  a  gate. 
Its  motion  is  circumferential 
instead  of  parallel  to  the  shaft, 
the  slots  and  intervening  solid 
portions  being  of  such  dimensions, 
in  connection  with  guide-vanes  of 
considerable  thickness,  that  while 
in  one  position  the  passage  of  the 
water  is  entirely  obstructed,  a 
comparatively  small  angular  mo- 
tion will  leave  the  guide-passages 
fully  open.  The  register-gate  is 
used  with  several  turbines  of 
American  make. 

Another  prominent  make  of  turbine  in  America  is  the 
"Victor  Turbine/'  now  (1905)  manufactured  .by  the  Platt 
Iron-works  Co.,  of  Dayton,  Ohio.  Fig.  68  gives  a  view  of 
the  turbine  itself,  with  its  peculiar  scooped-shaped  vanes; 
while  in  Fig.  69  is  shown  the  outer  wheel-case,  and  guides,  of 
one  of  these  turbines,  with  its  shaft  directly  connected  to  that 
of  an  electric  generator  vertically  above.  The  engraving  also 
shows  the*  steel  casing  forming  the  lower  terminus  of  the  flume  or 
penstock  conducting  water  to  the  turbine,  which  is  a  "  33-inch 


FIG.  67a. 


FIG.  68.     Victor  Cylinder  Gate  Runner. 


FIG.  69.     Victor  Turbine  in  Flume,  Direct-connected 
to  Generator. 


§  86.  AMERICAN   TURBINES.  133 

Cylinder-gate  Victor  Turbine/'  In  Fig.  70*  is  shown  the 
""Victor  High-pressure  Runner"  intended  for  heads  of  from 
100  to  2000  ft.  All  of  these  wheels  are  cast  in  one  piece,  of 
cast  iron  or  bronze. 

The  "'New  American  Turbine  "  manufactured  (in  1905)  by 
the  Dayton  Globe  Iron-works,  of  Dayton,  Ohio,  is  a  prominent 
American  motor  of  the  "inward  and  down  ward "  type.  The 
'"'runner/'  or  turbine  itself,  is  shown  in  Fig.  Tl.f  Two  kinds 
of  gate  are  used  with  this  turbine :  the  cylinder-gate,  as  already 
described  in  connection  with  other  turbines,  moving  parallel 
to  the  shaft;  and  the  "wicket-gate,"  which  consists  in  having 
.a  movable  leaf  on  one  side  of  each  guide-channel,  this  leaf 
being  pivoted  at  the  extremity  nearest  the  turbine  and  pro- 
vided with  a  rounded  shoulder  at  the  other.  By  the  swinging 
•of  this  leaf  the  waterway  of  each  guide-channel  is  varied  in 
amount,  and  may  be  closed  entirely. 

Fig.  72,  which  gives  a  horizontal  section  made  through 
the  upper  part  of  a  New  American  Turbine  (as  made  in  1890) 
.and  its  guides,  shows  these  movable  blades  or  leaves,  this 
arrangement  of  regulation  being  somewhat  similar  to  that 
adopted  in  the  Thomson  Vortex  Wheel  (see  §  80).  In  Fig.  73 
is  given  a  view  of  the  wheel-case  and  shaft  of  a  "'wicket-gate'7 
New  American  Turbine,  set  in  the  floor  of  a  wooden  flume. 
The  small  shaft  on  the  right  is  for  moving  the  guide-leaves, 
•each  of  which  is  connected  by  an  outside  link  (visible  in  the 
figure)  with  a  horizontal  disc.  When  the  small  shaft  turns, 
the  disc  also  turns  and  moves  all  the  guide-leaves  simultaneously 
and  through  the  same  extent  of  movement.  As  seen  in  the 
figure  the  discharge-tube,  or  short  "draft-tube/'  as  it  may  be 
called,  has  its  lower  edge  somewhat  below  the  surface  of  the 
tail-water. 

Other  prominent  American  turbines  of  the  "mixed-flow" 
type,  like  those  just  described,  are  the  "Hercules/7  made  by 
the  Holyoke  Machine  Co.,  of  Holyoke,  Mass.;  the  McCormick, 
made  by  the  S.  Morgan  Smith  Co.,  at  York,  Pa.;  and  the 
"Samson/7  by  the  James  Leffel  Co.,  of  Springfield,  Ohio.  This 
last-named  wheel  is  shown  in  Fig.  74,  opp.  p.  134,  and  is  in 

*  See  opposite  p.  125.  t  See  opposite  p.  135. 


134  HYDRAULIC   MOTORS.  §  &/. 

reality  a  double  wheel,  the  upper  portions  of  the  wheel-pas- 
sages being  partitioned  off  as  shown.  The  first  two  wheels  use 
a  cylinder-gate,  moving  axially,  i.e.,  parallel  to  the  shaft. 

The  trade  circulars  of  some  of  the  makers  of  the  fore- 
going turbines  refer  to  tests  of  their  wheels  made  at  Holyoke, 
Mass.,  at  the  testing-flume  of  the  Holyoke  Water-power  Co. 
(see  §  97),  where  the  highest  head  available  is  18  ft.  Some  of 
the  values  of  efficiency  obtained  in  these  tests  not  only  greatly 
exceed  80  per  cent.,  but  in  some  cases  approach  90  or  over. 
Polishing  and  smoothing  of  the  surface  of  the  turbine-vanes 
has  been  found  to  increase  the  efficiency  in  several  instances. 
In  the  case  of  several  American  turbines  taken  to  Europe  and 
there  re-tested,  European  methods  being  followed,  somewhat 
lower  values  of  efficiency  have  resulted.  It  is  thought  that 
the  discrepancy  is  due  to  the  differing  modes  of  measuring 
the  water  used. 

The  Jonval  type  of  turbine,  or  "  parallel-flow"  variety,  is 
manufactured  by  R.  D.  Wood  and  Co.  of  Philadelphia,  Pa., 
and  is  sometimes  made  "duplex";  that  is,  the  runner  is  pro- 
vided with  two  concentric  rings,  each  containing  a  set  of  vanes, 
the  guide-ring  being  double  also.  When  the  supply  of  water 
is  reduced,  one  ring  alone  is  brought  into  action  without  sac- 
rifice of  efficiency. 

As  already  mentioned,  the  firm  of  Kilburn,  Lincoln  and 
Co.,  at  Fall  River,  Mass.,  manufacture  an  outward-flow  turbine 
of  the  Fourneyron  type.  See  §  75  and  Figs.  17,  18,  19. 

87.  American  Turbines.  Historical.  (See  paper  by  Mr. 
Samuel  Webber  in  Transac.  Am.  Soc.  M.  E.  for  1905,  abstracted 
in  the  Engineering  News  of  Dec.  5,  1895;  and  also  one  by 
Mr.  A.  C.  Rice,  published  in  the  Engineering  News  of  Sept.  18, 
1902,  p.  208.)— During  most  of  the  first  half  of  the  nineteenth 
century  the  large  mills  of  New  England  made  use  of  the  over- 
shot and  breast  wheel  for  water-power;  but  in  1844  Mr.  Uriah 
A.  Boyden  built  and  installed  a  Fourneyron  turbine  of  75  H.P. 
at  Lowell,  Mass.,  which  on  test  yielded  an  efficiency  of  78 
per  cent.,  a  figure  considerably  greater  than  that  furnished 
by  the  old-fashioned  wheels  in  the  neighboring  factories. 


FIG.  74.     The  Samson  Runner. 


FiG.  72.     A  type  of  Wicket  Gate. 


i. 


FIG.  71.     New  American  Runner. 


§  87.  AMERICAN   TURBINES.  135 

Several  other  turbines  were  then  built,  some  of  which  were 
tested  by  Mr.  J.  B.  Francis,  hydraulic  engineer,  and  the  success 
of  these  motors  stimulated  imitation  and  invention  in  the 
United  States;  and  turbines  of  the  inward-flow  (Francis)  and. 
parallel-flow  (Jonval)  types  were  constructed  and  put  into 
service  in  many  mills.  About  1860  and  later,  the  Swain  and 
Leffel  turbines  were  invented,  combining  the  features  of  the; 
Francis  and  Jonval  types  by  securing  an  "  inward  and  down- 
ward "  discharge  of  the  water  (mixed  type).  The  dimensions 
of  the  wheel -passages  parallel  to  the  shaft  being  made  relatively 
great,  a  given  quantity  of  water  could  be  used  with  a  less 
diameter  of  wheel;  while  the  angular  velocity  (or  revolutions 
per  minute)  was  greater  for  a  given  linear  velocity  of  the  outer 
edge  of  wheel;  that  is,  for  a  given  head.  These  features  con- 
duced to  cheapness  of  construction  and  speed  in  operation. 
High  efficiencies  were  also  obtained  with  these  turbines;  those 
at  "part  gate "  being  a  notable  improvement  on  previous 
results.  The  Leffel  wheel  was  provided  with  the  "  wicket- 
gate  "  device  (as  at  present  in  the  " Samson "  and  "New  Ameri- 
can"). The  Swain  wheel  had  a  form  of  gate  which  main- 
tained a  rounded  aperture  at  all  stages. 

To  quote  from  Mr.  Webber's  paper:  "The  Swain  wheel 
had,  however,  given  an  excellent  result  as  far  back  as  1862, 
and  from  that  date  down  to  about  1878  the  number  of  turbines 
was  legion,  in  all  sorts  of  variations  of  curve  of  bucket  and 
form  of  gate,  but  all  containing  the  same  general  features  of 
inward  and  downward  discharge."  "The  general  result  of. 
this  change  from  the  Fourneyron  type,  as  first  introduced,, 
has  been  to  furnish  the  public  with  turbines  of  equal  power, 
in  one-half  the  space  and  at  one-fifth  the  cost,  being  single 
castings  of  iron  or  bronze  instead  of  being  built  up  of  many 
parts." 

In  1876  began  the  "new  departure"  in  the  design  of  Amer- 
ican turbines,  inaugurated  by  the  high  efficiency  at  part 
gate,  and  large  capacity  for  its  diameter,  of  a  24-in.  "Her- 
cules" wheel  invented  by  Mr.  John  B.  McCormick.  The  axial 
dimensions  of  this  wheel  were,  relatively,  greater  than  ever 


136  HYDRAULIC    MOTORS.  §  88. 

before,  and  each  bucket  was  provided  with  three  sharp  projecting 

ridges  to  assist  in  the  guidance  of 
the  water  at  "'part  gate."  (The 
"Hercules  "  of  that  date  is  shown 
in  Fig.  72a.)  Other  makers  soon 
followed  with  improved  designs  of 
their  wheels,  there  being  thus 
produced  the  "Victor,"  "'Risdon," 
and  "New  American";  all  with 
high  efficiencies  and  large  capac- 
ity for  a  given  diameter.  Mr. 

Webber  gives  a  table  showing  the  progressive  increase  in 
capacity  (for  a  given  diameter)  from  the  Boyden-Fourneyron 
design,  with  which,  in  the  case  of  a  36-in.  wheel  under  26  ft. 
head,  22.95  cub.  ft.  of  water  was  used  per  second,  up  to  the 
more  recent  "Samson"  "Hercules"  and  "Victor"  wheels,  each 
of  36  in.  diameter  and  using  109  cub.  ft.  of  water  per  second 
under  the  same  head,  26  ft.,  and  with  even  greater  efficiency. 
The  "  Hercule-Progres"  made  in  France  on  an  American  type, 
has  the  same  general  appearance  as  the  "Hercules"  shown  in 
Fig.  72a.  (See  PrasiPs  Report  on  Turbines  at  the  Paris  Expo- 
sition of  1900;  Schweizerische  Bauzeitung,  vols.  36,  37.) 

88.  Choice  of  Hydraulic  Motor  for  Different  Heads. — A 
valuable  article  by  Mr.  John  Wolf  Thu  so* on  "Modern  Turbine 
Practice  and  the  Development  of  Water- powers"  was  published 
in  the  Engineering  News  of  Dec.  4, 1902,  p.  46,  and  Jan.  8,  1903, 
pj.  26.  The  following  recommendations  are  quoted  from  that 
article : 

"The  type  (of  hydraulic  motor)  to  be  employed  in  each 
individual  case  should  be  in  accordance  with  the  height  of  the 
head  to  be  utilized,  as  follows: 

1.  "Low  heads,  say  up  to  40  ft.:  American  type  of  turbine 
(i.e.,  of  the  "inward  and  downward"  variety)  on  horizontal 
or  vertical  shaft,  in  open  flume  or  case,  nearly  always  with  draft- 
tube. 

2.  "Medium  heads,  say  from  40   to   300   or  400  ft. :   Ra- 
dial inward-flow  reaction  or  Francis  turbine,  with  horizontal 

*  See  also  Mr.  Thurso's  book  mentioned  in  the  "  .bibliography"'  on  p.  iv. 


FIG.  73.     New  American  Turbine  in  Wood  Flume. 


89. 


AMERICAN    TURBINES. 


137 


shaft  and  concentric  or  spiral  cast-iron  case  with  draft- 
tube. 

3.  "High  heads,  say  above  300  or  400  ft.:  Pelton  wheel; 
or  radial  outward-flow,  segmental-feed,  free -deviation  turbine 
(i.e.,  a  Girard  impulse  wheel);  or  a  combination  of  both,  on 
horizontal  shaft  and  cast-  or  wrought-iron  case;  often  with 
draft-tube/' 

89.  General  Theory  of  Reaction  Turbines. — The  theory  of 
the  mixed-flow  turbine  will  now  be  presented  and  finally 
generalized  so  as  to  be  applicable  to  any  type  of  reaction  tur- 
bine. Fig.  75  represents  a  single  passageway,  1  .  .  .  N,  of  a 


t '' 


FIG.  75. 

mixed-flow  turbine  having  its  shaft,  S,  vertical.  The  entrance- 
point,  1,  is  describing  a  horizontal  circle  A  ...  1 ...  M  with 
a  velocity,  Vi,  of  proper  value  for  best  effect,  the  corresponding 
velocity  of  the  exit-point  N  being  vn,  in  horizontal  circle 
T  .  .  .  N .  . .  G,  this  circle  being  a  vertical  distance,  h0,  below 
position  1,  which  is  itself  hi  ft.  below  the  surface  of  the  head- 
water. The  wheel  is  supposed  to  be  in  an  open  flume  or  wheel- 
pit,  so  that  there  is  no  loss  of  head  in  a  penstock;  in  fact  all 
friction  in  guide-passages  and  wheel-channels  will  be  disre- 
garded, at  first.  There  is  no  diffuser,  so  that  the  fluid  pressure 
of  the  water  at  the  point  of  exit  N  is  taken  as  pn  = 


138  HYDRAULIC    MOTORS.  §  89, 

b  being  the  height  of  the  water-barometer  and  hn  the  vertical 
distance  of  the  point  N  below  the  surface  of  the  tail-water. 
(In  case  N  is  above  the  tail-water,  hn  is  negative.)  The  paral- 
lelogram of  velocities  at  entrance  is  in  a  horizontal  plane,  Wi 
being  the  absolute  velocity  of  the  water  at  that  point  making 
an  angle  a  with  wheel-rim  tangent,  and  c\  the  relative  velocity 
It  will  be  assumed,  as  before,  that  the  angle  /?  is  eventually  to 
have  such  a  value  that  there  will  be  no  "shock,"  or  impact, 
at  entrance.  .  At  the  exit-point  N  the  parallelogram  of  velocities 
is  in  a  vertical  plane,  the  absolute  velocity  wn  being  the  diagonal 
formed  on  the  relative  velocity  cn,  as  one  side,  and  the  wheel- 
rim  velocity  vn,  as  the  other  side,  of  a  parallelogram.  Of 
course  the  relative  velocity  cn  follows  the  tangent  to  the  walls 
of  the  turbine  passage  at  N  and  makes  a  (small)  angle  d  with 
the  wheel-rim  tangent  Nvn>  The  two  radii  are  r\  and  rnj  as 
shown,  rn  being  the  radius  of  the  mean  position  N,  or  point 
half-way  out  radially  along  the  discharging  edge  (shown  better 
in  the  next  figure,  Fig.  76)  .  Let  us  denote  by  Fn  the  aggregate 
sectional  area  of  the  exit  passages  of  the  turbine,  that  of  each 
passage  being  taken  at  right  angles  to  the  relative  velocity  cn; 
and  by  F0  the  aggregate  sectional  area  of  the  guide-passages 
at  the  entrance-point,  1.  For  example,  in  Fig.  76,  if  there 
are  mn  turbine  channels  and  m0  guide-passages,  then  Fn  = 
mnanen  and  F0  =  m0ae0  sq.  ft. 

We  now  have  the  following  relations  (losses  of  head  in 
guide-passages  and  wheel-channels  being  neglected): 

From  .trigonometry, 

Ci2  =  Wi2+Vi2  —  2wiVi  cosa,     .     .     .     .     (1) 
and  wn2  =  cn2+vn2-2cnvncosd  .....     (2) 

From  Bernoulli's  Theorem  applied  to  the  steady  flow  of 
the  water  between  rigid  stationary  walls  from  head-water 
surface  to  outlet  of  guides, 


(pi  being  the  internal  fluid  pressure  at  point  1). 


§  89.  TURBINES.       GENERAL    THEORY.  139 

From  Bernoulli's  Theorem  for  a  steady  flow  in  a  rigid  pipe 
rotating  uniformly  about  a  vertical  axis  (see  eq.  (13),  §  41) 
between  entrance-point  1  and  exit-point  N  of  a  turbine  channel 
(adding  in  the  gravity  head  /i0), 


For  a  minimum  residual  kinetic  energy  we  may  write,  as 
in  previous  investigations,  the  relative  velocity  cn  =  wheel-rim 
velocity  vn  at  exit,  i.e., 

cn  =  i?«  (see  §  53)  .......  (5) 

Equation  of  continuity:    F0Wi=Fncn;         ......  (6) 

The  proportion:  Vi:vn:  :ri\rn:    .......  (7) 

The  volume  of  water  used  per  second: 

Q=FQwlf  =  Fncn  ......     .  (8) 

Theoretic  power  of  the  wheel,  in  case  there  is  no  diffuser  and 
all  fluid  friction  between  head-water  surface  and  point  of 
exit  N  (also  axle-friction)  be  neglected,  is 

Or  w  2 

L,=R'v',=Qrh-^-£-;   jft.-lbs.persec.},   .     .     (9) 

<J 

Rf  being  the  resistance,  Ibs.  (overcome  by  the  turbine  in  steady 
running),  tangent  to  the  circumference  of  a  pulley  where  the 
linear  velocity  is  vf  ft.  per  sec.  (N.B.  If  by  means  of  a  dif- 
fuser all  loss  of  head  between  exit-point  N  and  the  surface  of 
the  tail-water  could  be  considered  to  be  obviated,  we  should 
have  R'v'=Qrh  as  in  eq.  (4),  §  38.) 

Now  solve  (3)  for  pi-^-f  an<^  substitute  in  (4),  from  which, 
after  inserting  the  value  of  ci2  from  (1)  and  noting  that 
hi+hQ—hn  =  h,  the  total  head  of  the  mill-site  (that  is,  the  ver- 
tical distance  from  the  surface  of  the  head-water  to  that  of 
the  tail-  water,  and  also  writing  cn=vn[hom  (5)],  we  have 

cosa=gh  .......     (10) 


.•  -,   x-N  Fncn      Fnvn  n 

But,  from  (6)  and  (7),  Wi=~^-,  =-&—;    and  vi  =  --  ;  sub- 

^0  ^0  >n 

stituting  which  in   (10),   and  solving,  we  have  as  the  "'best 


140  HYDRAULIC    MOTORS.  §  90. 

value"  of  the  exit  wheel-rim  velocity  for  best  effect  when  fluid 
friction  is  disregarded  (with  the  exception  of  that  between 
point  N  and  surface  of  tail-water,  there  being  no  diffuser) 


cos  a 


This  is  now  in  such  a  form  as  to  hold  good  for  any  reaction- 
turbine,  the  subscripts  1  and  n  referring  to  entrance  and  exit, 
respectively,  of  the  turbine  channels;  and  a  fair  allowance 
for  fluid  friction  in  the  guide-passages  and  turbine  channels 
may  be  made  (as  due  to  a  study  of  numerous  numerical  examples 
and  actual  tests)  by  deducting  eight  per  cent,  of  this  value 
from  itself;  that  is,  by  writing* 


In  the  case  of  a  parallel-flow,  or  "axial,"  turbine,  r\=rn 
(  =  r),  being  measured  to  the  middle  point  of  the  radial  dimen- 
sion of  the  ring  containing  the  wheel-vanes;  see  Fig.  62. 

go.  Turbines.     General  Theory  with   Friction.  —  If   in  the 

analysis  of  the  last  paragraph  we  introduce  a  loss  of  head  ^~- 

between  head-water  and  entrance-point  1,  and  a  loss  of  head 

cn2  . 
£n7T~  in  the  turbine  channels  themselves,  with  cn  =  vn  as  before, 

for  best  effect,  the  outcome  is  found  to  be 


Vn=    I — rTT ~  ~r FV NF'n'SoBa-    (13) 

\l  i     I   sO       -*•   n  'n  L,n  L   0'  n  -1   n    '  1    *^Uk 

0    ^  -     cos  a      2    Fn7*i  cos  a 


For  ordinary  values  of  the  ratios  of  the  radii  and  sectional 
areas  concerned,  and  with  £o  and  £n  each  equal  to  about  0.10, 
.as  mentioned  in  §  71,  the  value  of  the  first  radical  in  eq.  (13) 
above  would  be  found  to  be  not  far  from  the  0.92  of  eq.  (12). 

It  is  to  be  noted  that  the  relation  w iVi  cos  a  =  gh,  in  eq.  (10); 
may  also  be  derived  in  a  much  more  direct  manner  by  the 
analysis  already  given  in  §  67,  which  applies  to  any  turbine 

*  A  different  form,  which  may  replace  eq.  (12),  is  arrived  at  in  the  ad- 
dendum on  p.  148. 


§  91.  TURBINES.   GENERAL  THEORY.  141 

whatever  since  the  parallelograms  of  velocities  at  entrance  and 
exit  may  lie  in  any  position  relatively  to  the  shaft  of  the  tur- 
bine without  vitiating  any  of  the  steps  taken  in  the  analysis 
of  that  paragraph.  See  §  68. 

Another  point  to  be  noted  is  that  eqs.  (1),  (2),  (5),  (6),  (7), 
and  (8)  apply  even  when  fluid  friction  in  guides  and  channels 
is  considered,  and  will  therefore  be  used  in  subsequent  opera- 
tions where  it  is  desired  to  take  account  of  that  friction. 

91.  Sectional  Areas  F0  and  Fn.  Thickness  and  Number  of 
Guide-blades  and  Turbine-vanes.  —  Let  us  consider  the  sectional 
area  of  the  cross-section  of  a  guide-passage  at  point  1  to  be 
rectangular.  It  is  perpendicular  to  the  absolute  velocity  w\, 
has  a  width  a  (  =md0  in  Fig.  47),  and  a  length  e0.  See  Fig. 
76,  in  which  is  also  shown  the  rectangular  section  of  a 
turbine  channel  at  exit;  likewise  considered  rectangular,  with 
width  an  and  length  en]  SS  is  the  shaft  of  turbine.  Now 
let  so  denote  the  "pitch"  of  a  guide-blade;  that  is,  the  length 
of  that  portion  of  the  circumference,  of  radius  r\,  which  corre- 
sponds to  one  guide;  also  let  ra0  be  the  number  of  guides  (so 
that  raoso=27rri)  and  t0  the  thickness  of  the  guide-blade. 
Similarly,  at  the  exit-point,  N,  of  a  turbine  channel,  let  sn  be 
the  pitch  of  a  turbine-vane,  mn  the  number  of  vanes,  and  tn 
the  thickness  of  a  vane.  Then  a  =  s0sma—tQ  (very  nearly), 
and  therefore,  since  Fo  =  moaeo,  we  have 

F0  =  e0(m0so  sin  a  —  ra0£o)  =  e0(2^ri  sin  a  —  7n0^o)  ;      -     (14) 

and  similarly 

Fn=en(2icrnzmd-mntn).  •    .     .     .     .     (15) 

For  approximate  purposes,  however,  we  may  write  the  ratio 


o_o  I 

Fn~en'smd'rn' 

The  above  is  readily  understood  for  radial  turbines.  As  for 
parallel-flow  wheels  (or  "axial  wheels")  we  write  ri=rn  (call 
it  now  r)  and  measure  it  to  a  point  half-way  between  the  inner 
and  outer  rings  between  which  the  turbine-vanes  are  placed. 
The  pitch  of  the  guides  and  vanes  is  measured  along  this  inter- 


142 


HYDRAULIC   MOTORS. 


§92. 


mediate  circle  of  radius  r,  and  the  dimensions  e0  and  en  are  radial 
(and  horizontal,  if  the  shaft  is  vertical;  see  Figs.  62  and  64). 

In  a  mixed-flow  turbine  (see  Fig.  76),  the  entrance  rectangle 
(that  is,  for  a  guide-exit),  of  area  =  ae0,  is  vertical  (provided  the 

^     J^  shaft  of  turbine  is  verti- 

.^— • — > — IT cal)>  while  at  the  turbine- 
exit,  Ar,  the  right  section 
between  vanes  has  two 
horizontal  edges  of  length 
=en  and  an  area=anen; 
where  rn  is  the  radius  of 
the  point  on  edge  of 
wheel -vane  at  A,rialf-way 
(say)  between  the  ex- 
treme points  of  that  hori- 
zontal edge.  en  is  hori- 
zontal and  may  be  greater  than  eo  if  desired.  (Of  course,  in 
the  case  of  turbine-vanes  which  have  scoop-shaped  forms  at 
the  exit -point  N,  the  dimensions  of  this  equivalent  rectangle, 
of  sides  en  and  ant  are  difficult  to  estimate.) 

92.  Empirical  Relations  for  Turbine  Design. — In  the  design 
of  a  particular  turbine  which  is  to  work  under  a  given  head  and 
utilize  a  given  quantity,  Q,  of  water  per  second,  there  are 
many  dimensions  and  quantities  which  have  to  be  determined 
and  finally  so  adjusted  to  each  other  as  to  produce  the  best 
results.  Theory  cannot  determine  all  these  quantities.  If  the 
guide-blades  and  wheel-vanes  are  too  numerous,  a  dispropor- 
tionate amount  of  rubbing  surface  is  offered  to  the  water, 
with  consequent  loss  of  power  from  fluid  friction;  whereas  if 
they  are  too  few,  the  water  is  not  so  well  guided  and  leaves  the 
wheel  with  too  great  absolute  velocity.  Durability  and  ease 
of  construction  are  also  to  be  regarded.  Experience  and  ex- 
periment, therefore,  must  be  relied  upon  to  a  large  extent  as 
governing  certain  elements  of  design.  In  America  the  evolu- 
tion of  the  "  inward  and  downward  "  turbine  has  been  very 
largely  a  matter  of  "cut  and  try";  but  very  good  results  have 
finally  been  attained  for  moderate  heads  (up  to  about  40  ft.). 


§  92.  TURBINES.      EMPIRICAL    RULES.  143 

In  Europe,  however,  it  is  more  generally  the  custom  to  design 
each  proposed  turbine  for  its  special  site  and  work. 

In  the  present  book  space  cannot  be  given  to  special  details 
of  design  and  construction.  For  these  the  reader  is  referred 
to  the  works  of  Bodmer  and  Thurso,  in  English;  and  to  those 
of  Meissner,  Mueller,  Herrmann,  and  Zeuner,  in  German. 

It  will  suffice  to  give  a  few  empirical  relations  and  assump- 
tions condensed  mainly  from  Bodmer  and  Mueller. 

If  we  distinguish  the  following  three  cases,  viz.: 

Case  I.  When  Q-Z-WI  is  >16  sq.  ft.  (large  Q  and  small  ft); 

Case  II.  When  Q+WI  is  >2  and  <16  sq.  ft.  (medium  Q 
and  ft); 

Case  III.  When  Q+Wi  is  <2  sq.  ft.  (small  Q  and  high  ft); 
then  the  angle  a  may  be  assumed  from  20°  to  25°  for  (I) ;  15° 
to  20°  for  (II);  and  15°  to  17°  for  (III);  for  axial  turbines. 

Angle  d  for  the  three  cases,  for  axial  turbines: 
20°  to  25°,     15°  to  17°,     and     12°  to  16°,  respectively. 

For  radial  inward-flow  and  mixed-flow  turbines  take  a  from 
10°  to  24°;  also  d  from  16°  to  24°. 

For  radial  outward-flow  turbines  assume  a  from  15°  to  24°, 
and  o  from  10°  to  20°. 

P 
The  ratio  ^~  for  axial  turbines  may  be  taken  as  0.5  to  1.5, 

*   n 

usually  =  1.0. 

For  radial  inward-flow  and  for  mixed-flow  wheels  take  ri  from 
0.75\7JF^to  1.75V^;  with  rn  =  0.65  to  0.85  of  n. 

For  radial  outward-flow  turbines  assume  7*1  from  1.50VF0 
to  2.0VJ^;  with  rn  =  1.25ri  to  l.SOri. 

For  axial  wheels,  referring  again  to  the  above  three  cases,  I, 
II,  and  III,  the  radius  r  may  be  taken  from  VF0  to  1.25\  f^ 
for  Case  I;  1.25V^  to  1.5V "f^  for  II;  and  1.5v  J^to  2v  Y~Q 
for  III;  also,  as  to  the  pitch,  take  s0=10  to  12  inches  for 
€ase  I;  from  r-^  3.75  to  r^-4.5  for  II;  and  4.5  to  6.0  inches 
for  III. 

For  radial  inward-flow  and  for  mixed-flow  turbines  the  pitch 
may  be  taken  from  4.5  to  12  inches;  and  for  radial  outward- 
flow  wheels,  from  r^4.5  to  r  +  6. 


144  HYDRAULIC   MOTORS.  §  93. 

.As  regards  the  value  of  en  with  axial  wheels,  Mr.  Boclmer 
recommends  a  value  of  from  r-5-  1.25  to  r-r-  2  in  Case  I  (above); 
from  r-^2  to  r-^-2.5  for  Case  II;  and  from  r-^2.5  to  rn-3  in 
Case  III.  Also,  for  the  axial  depth  of  wheel  in  the  case  of  an 
axial  turbine,  from  r-^5  to  r-^3. 

The  number  of  turbine-  vanes,  viz.,  mn  (  =  27rrn-^sn),  should 
be  greater  by  1  or  2  than  the  number,  m0,  of  guide-blades. 
The  thickness  of  both  guide-blades  and  turbine-vanes  should 
be  taken  at  from  J  to  f  inch  for  cast  iron,  and  from  J  to  f  inch 
for  wrought  iron  or  steel;  these  dimensions  for  parts  near  the 
ends,  which  should  be  beveled  or  sharpened  off  if  possible. 
(It  may  sometimes  be  advantageous  to  make  the  wheel  vanes 
much  thicker  in  their  middles  to  give  better  form  to  the 
passageways  between  them:  see  Fig.  55.) 

93.  Computations  for  a  Proposed  Turbine.  Order  of  Pro- 
cedure. —  It  is  supposed  that  the  rate  at  which  water  may  be 
used  in  steady  flow,  viz.,  Q  cub.  ft.  per  second,  is  given  and  also- 
the  head  h  of  the  mill-site,  and  that  a  turbine  of  some  special 
type  is  to  be  designed  for  the  given  site  and  duty.  We  should 
first  assume*  values  for  the  two  ratios  Fo+Fn  and  rnH-r1;  and 
for  the  angle  a.  These  assumptions  may  need  to  be  revised, 
however,  after  a  certain  progress  has  "been  made  with  the 
computations.  For  example,  with  a  radial  turbine,  to  have 
the  crown-plates  parallel  implies  equal  values  for  e0  and  en,  in 
which  case  the  assumption  of  the  three  values,  Fo+Fn,  rn  +  rir 
and  a  would  determine  a  value  for  the  angle  <5;  which  value 
might  not  be  suitable,  thus  requiring  a  change  in  some  of  the 
original  assumptions.  In  such  a  case,  therefore,  (radial  turbine 
with  parallel  crowns,)  it  would  be  better  to  assume  the  three 
values  a,  d,  and  rn-*-ri;  from  which  the  ratio  F0+Fn  could  then 
be  computed  (at  least  approximately)  from  eq.  (16)  of  §  91. 

With  these  three  values,  then,  viz.,  for  Fo+-Fn,  rn^rlr 
and  a,  —  h  being  already  given,  —  we  find  the  best  speed  for  the 
exit  wheel-rim,  vn,  from  eo.  (12),  §  89,  viz., 


(in  deriving  which  fluid  friction  has  been  taken  into  account). 

*  Or,  we  may  assume  rn  +  rv  a,  and  ft;   and  then  follow  the  method  of 
the  addendum  on  p,  118. 


§  94.  TURBINES.      GENERAL   THEORY.  145 

With  vn  known  we  now  apply  the  relations  vn=cn  for  best 
effect  and  Fncn  =  F$WI  (both  of  which  hold  even  when  friction 
is  considered),  and  solve  for  wif  the  absolute  velocity  at  entrance. 

We  are  now  in  a  position  to  determine  the  quotient  Q+  w\t 
upon  which  the  choice  of  angles  a  and  d  partly  depends  for 
axial  turbines.  If  necessary,  a  new  choice  may  now  be  made 
and  the  computation  revised. 

With  Wi  and  v\  known  [since  Vi=  (ri-^rn)vn],  the  parallelo- 
gram of  velocities  at  the  point  of  entrance  of  the  turbine  channel 
can  be  solved,  trigonometrically  or  graphically,  and  thus  the 
value  of  the  angle  /?  becomes  known,  upon  which  depends  the 
position  of  the  relative  velocity  c\  and  of  the  tangent  to  wheel- 
vane  at  this  entrance-point  (see  Figs.  48,  58,  and  62).  The 
tangent  to  the  wheel-vane  at  1  must  have  this  position  in 
order  that,  at  the  speed  of  wheel  juct  previously  found  (vi), 
there  may,  at  no  "shock"  or  impact  at  point  1.  In  Mueller's 
recent  work*  the  statement  is  made  that  recent  practice  in 
designing  Francis  turbines  favors  the  relations  Wi  =  Vgh  and 
/?=  about  90°. 

From  the  now  known  value  of  FQ,  =  Q-*-w\,  we  pass  on  to 
the  computation  of  the  value  of  the  radius  n  from  the  empirical 
rules  of  §  92;  from  which  follows  that  of  rnj  since  the  ratio  of 
the  radii  was  assumed  at  the  outset.  The  pitch  of  the  guide- 
blades  is  then  fixed  upon  from  the  rules  given  in  the  preceding 
paragraph  and  the  number  of  guide-blades  computed,  i.e., 
T??O.  The  value  of  e0  is  now  found  from 

Fo  =  C[27cri  sin  a- m0to]e0,      .     .j    .     .     (17) 

in  which,  according  to  Mr.  Bodmer,  the  value  of  C  is  to  be 
taken  as  1.0  for  axial  wheels,  and  as  0.91  for  radial  and  mixed- 
flow  wheels. 

94.  Excess  Pressure  at  Entrance. — In  German  treatises  on 
turbines  some  stress  is  laid  on  the  desirability  of  adopting 
finally  such  dimensions  and  angles  that  the  fluid  pressure  at 
point  1  (entrance)  shall  not  greatly  exceed  that  in  the  tail- 

*  "Dice  Francis-Turbinen  und  die  Entwicklung  des  Modernen  Turbinen- 
baues,"  by  W.  Mueller.     Hannover,  1901. 

10 


146. 


HYDRAULIC    MOTORS. 


§94a. 


water  or  draft-tube  on  the  other  side  of  the  joint  or  crevice 
formed  between  the  edges  of  the  turbine  crowns  and  the  oppo- 
site, adjoining,  stationary  edges  of  the  guide  floor,  or  gate. 
If  the  difference  is  great,  the  leakage  through  the  crevice  may 
be  of  importance.  The  amount  of  the  (unit)  pressure,  pi, 
may  be  found  from  eq.  (3),  §  89,  after  wi  has  been  determined. 

94a.  Numerical  Example.  Jonval  Turbine.  Fig.  62. — Com- 
pute proper  values  for  the  arrangement  of  a  parallel-flow  tur- 
bine (Jonval)  which  is  to  work  under  a  head  of  h  =13.5  ft. 
and  to  use  Q  =  210  cub.  ft.  per  sec.  of  water. 

Here  let  us  assume  the  angles  a  and  d  to  be  20°  and  15° 
respectively,  and  that  e0  =  en  (nearly;  for  present  purposes,  in 
order  to  compute  the  ratio  Fo  +  Fn)-  Since  ri  =  rn,  =  r,  the 

ratio  rn+Ti  is  unity;  therefore,  from  eq.  (16),  §  91,  TT  = 

j?  n    sin  o 

0.342 


0.259 


1.32;  and,  from  eq.  (12), 


x- 


32.2X13.5 


0.940 


22.72  ft.  per  sec. 


Next, 


17,2  ft.  per 


To  find  the  angle  /?  of  Fig.  62,  note  that  in  Fig.  77,  showing 
the  parallelogram  of   velocities   at   entrance,   if   the   perpen- 


FIG.  77. 


dicular  FE  be_dropped_jroin  F  upon  1.  .D,  wejhave  DE,  or 
171?,  =Dl-El]_i.e.2_pE  =  v1-w1  cos  a.  Also,  FE=Wi  sin  a. 
Now  the  ratio  FE+  DE  =  tsn.  /?',  /?'  being  the  supplement  of  the 


§  94a.  TURBINES.       NUMERICAL   EXAMPLE.  147 

desired  angle  /?;  hence  (remembering  that  vi  =  vn  for  this  type 
of  wheel)  we  have 

wl  sin  a  17.2X0.342 


n-uncosa~22.72-17.2xO.94Q 


_ 
~ 


Therefore  f  =  43°  40'  and/?  =  136°  20'.    F0  =  ^  =  ^  =  12.2  sq.  ft. 

and  hence,  assuming  r=  1.25V  J^  (see  §92),  r=  1.25V  12.2  = 
4.36  ft.;  i.e.,  the  mean  diameter  of  the  turbine  should  be 
2r=8.72  ft.  Adopting  36  as  the  number  of  guide-blades  and 
38  wheel-vanes,  with  the  thickness  t0=tn  =  Q.5  in.  near  ex- 
tremities, we  have  for  the  length  of  opening  of  a  wheel-channel 
at  entrance  [radial  in  position;  see  Fig.  62,  and  eq.  (17),  §  93] 

FQ  _  12.2  _ 

60  ~  2nr  sin  a  -  ra0fo     2;r  X  4.36  X  0.342  -  36  X  & 

=  12.2-  7.86=  1.55  ft.,  =  1  ft.  6.6  in. 

If  the  thickness  of  blades  and  vanes  were  zero,  e0  and  en  would 
be  equal,  since  the  ratio  F0  +  Fn  has  been  taken  equal  to 
sin  a  -i-  sin  d.  But,  taking  into  account  the  thickness  (0.5  in.), 
we  find  for  en,  from  eq.  (15),  §  91, 

Fn  ___  12.2-1.32 
€n~27tr  sin  3-mntn~2nX  4.36(0.259)-  38  X^V 
=  1.63  ft.;   which  is  so  little  different  from  e0  that  the  work 
of  determining  the  best  speed  vn  (which  assumed  eo  =  en)  need 
not  be  recast.     (Or,  the  thickness  of  vanes,  tn,  at  exit,  might  be 
made  a  little  smaller  than  that,  t0,  of  guides  near  entrance  of 
wheel,  in  the  proportion  of  sin  d  to  sin  a.     In  that  case  en 
would  be  more  nearly  equal  to  e0.) 

The  final  dimensions  of  the  turbine,  then,  are: 
Outer  diameter  =  2r  +  e0  =  10.27  ft. 
Inner  diameter  =  2r  —  e0=  7.17ft. 

The  depth  of  the  wheel  (  =  Sl)  in  Fig.  62)  may  be  taken 
as  r-r-  5;  that  is,  as  0.87  ft. 

The  probable  efficiency  to  be  expected  is  some  80  per  cent. 
or  over  (full  gate),  this  being  about  the  figure  reached  (83  per 


148  HYDRAULIC  MOTORS.  §  94a. 

cent.)  in  the  test  of  a  turbine,  closely  resemKing  that  of  the 
present  example,  at  Goeggingen,  Germany  (see  Bodmer's  Hy- 
draulic Motors,  p.  365),  at  its  best  speed  of  45.5  revs,  per  min, 
(and  full  gate). 

In  the  present  example,  from  the  value  of  the  best  linear 
speed,  vn  =  22.72  ft.  per  second,  of  the  extremity  of  the  mean 
radius,  we  compute  the  angular  speed  as  follows,  if  n  denote  the 
revs,  per  unit  time: 


or, 
n  =  22.72  -*-  (2;rX  4.36)  =0.83  revs,  per  sec.,   =49.8  revs,  per  min, 

Addendum  to  §§89  and  93.  —  (This  procedure  may  be  sub- 
stituted for  that  of  §  93.)  From  the  parallelogram  of  velocities 
at  the  entrance  point,  1,  (see  Fig.  75)  we  have 

Wi'.Vi  ::  sin/?:  sin  (/?—«),     .....     (18) 
which,  in  eq.  (10),  p.  139,  leads  finally  to  the  relation 


Vi=Q         Usi 
\   sin 


/?  cos  a   ' 

to  replace  the  form  derived  as  eq.  (12),  p.  140. 

Hence  we  may  assume  a  and  /?  and  the  ratio  rn  -f-  r\  ;  and 
find  Vi  from  eq.  (19);  and  then  vn,  which  =  (rnH-ri)Vi;  also  w\ 
from  (18).  FQ  follows,  from  F0  =  Q  -:-  w{  ;  while  TI  is  chosen  by 
the  rules  of  §  92,  as  also  the  pitch  and  number  of  guide-blades. 
We  then  find  eQ  from  eq.  (17).  Having  substituted  Fn,  '=Q  + 
cn,  =  Q+Vn,  in  eq.  (15),  we  assume  en  and  compute  the  angle  d 
from  that  equation,  or  vice  versa;  but  the  limitations  for  this 
angle  d,  as  stated  on  p.  143,  must  be  observed.  If  necessary,  a 
revision  must  be  made  of  the  assumptions  for  a  and  /?.  (This 
method  is  used  by  Weisbach,  and  also  in  the  article  by  Prof. 
S.  J.  Zowski  in  the  Engineering  News  of  Jan.  6,  1910,  p.  20.  The 
angle  d,  however,  is  not  mentioned  in  this  article.) 


CHAPTER  VI. 
TESTING  AND  REGULATION  OF  TURBINES. 

95.  The  Prony  Friction-brake. — In  case  a  turbine  is  *  jed 
to  run  an  electric  generator  on  the  same  shaft,  its  power  at 
different  speeds  may  be  tested  by  electrical  measurements 
applied  to  the  electric  current  produced;  but  ordinarily,  if 
the  turbine  is  not  too  large,  use  is  made  of  a  Prony  friction" 
•brake  (see  p.  158,  M.  of  E.)  applied  to  the  rim  of  a  pulley  secured 
on  the  shaft  of  the  motor.  By  the  tightening  of  ih'i  brake 
more  or  less  friction  is  produced  on  the  pulley-rim,  and  the 
value  of  this  friction  becomes  known  by  the  weights  iecessary 
to  hold  the  brake  in  equilibrium;  that  is,  to  prevent  £he  brake 
from  being  turned  by  the  pulley  which  moves  wi,nin  it.  If 
the  pulley  has  a  horizontal  axle,  the  weights  an  suspended 
from  the  extremity  of  a  lever  projecting  from  tht  brake  and 
forming  a  rigid  part  thereof;  but  in  case  the  shaft  (A  the  pulley 
is  vertical,  the  rod,  or  scale-pan,  on  which  the  weights  are 
suspended  is  connected  with  the  rim  of  the  bral.6  by  a  bell- 
crank  lever. 

The  friction  thus  provided,  and  measured,  becomes  for  the 
time  being  the  o  ly  resistance  Rf  (Ibs.)  besides  the  axle  friction 
of  the  shaft  itself  The  linear  velocity,  v',  of  the  rim  of  the 
pulley  becomes  known  from  a  measurement  of  the  rate  of  rota- 
tion (revs,  per  minute)  and  the  radius  of  the  pulley- rim.  In 
any  one  experiment,  then,  the  power  R'v'  (ft.-lbs.  per  second) 
is  the  power  developed  by  the  turbine  over  and  above  that 
(fi'V)  spent  on  axle  friction.  If  the  brake  is  tightened  suffi- 
ciently, all  motion  on  the  part  of  the  turbine  may  be  prevented 

149 


150 


HYDRAULIC    MOTORS. 


96. 


H 


FIG.  79. 


and  the  power  is  zero,  since  vf  is  zero. 
On  the  other  hand,  if  no  resistance  of 
friction  or  otherwise  (except  axle  friction) 
is  provided  at  the  pulley -rim  or  else- 
where, then,  although  a  high  rate  of 
rotation  may  be  maintained  by  the  tur- 
bine when  thus  run  "  unloaded."  the  use- 
ful power  developed  is  again  zero;  since 
R'  is  zero.  Roughly  speaking,  it  may 
be  said  that  the  speed  of  steady  motion 
assumed  by  a  turbine  when  thus  run 
" unloaded"  is  about  double  that  to 
which  it  adjusts  itself  in  steady  motion 
when  a  resistance  R'  is  applied  of  such 
value  that  the  product  R'v',  or  useful 
power,  is  a  maximum. 

96.  The  Hook  Gauge. — A  useful  in- 
strument employed  for  the  determination 
of  the  position,  or  change  of  position,  of 
the  horizontal  surface  of  a  body  of  still 
water,  is  the  "hook  gauge."  If  a  vertical 
rod  of  metal  have  its 
lower  end  turned  up- 
ward in  the  form  of 
a  hook,  with  a  fine 
sharp  point,  as  in 
Fig.  78,  the  point  of 
the  hook  can  be  ad- 
justed to  the  level  of 
the  water  surface  with 
great  precision.  The 
instant  when,  in  its 
upward  motion,  the 
£EE  point  of  the  hook  is 

i^-t±^fi  i-i  just   about  to  break 

— —  "  through    the   surface 

FIGHTS."       ~  can    be    noted    with 


M 


N 


§  97.  TURBINE    TESTING.  151 

great  exactness  by  the  eye  of  the  observer,  if  the  water  is 
quiet;  and  the  upward  motion  of  the  stem  MN  may  then 
be.  arrested.  Fig.  79  shows  such  a  hook,  H,  attached  to  a 
graduated  rod  (like  a  leveling-rod)  supported  between  two 
fixed  vertical  guides  carrying  a  vernier  reading  to  thousandths 
of  a  foot.  A  nut,  N,  is  attached  to,  and  projects  from,  the  rod;, 
and  both  nut  and  rod  are  caused  to  travel  vertically  when  the- 
milled  head  A,  and  with  it  the  screw  G,  is  turned.  If  the 
vertical  distance  of  the  zero  of  the  vernier  above  the  sill  of  a 
weir  (for  instance)  is  known,  and  also  that  of  the  zero  of  the 
scale  above  the  point  of  the  hook,  the  vertical  height  of  the 
point  of  hook  above  the  sill  of  weir  at  any  time  is  easily  com- 
puted from  the  observed  reading  of  the  vernier  on  the  scale. 
The  instrument  in  Fig.  79  is  one  of  those  used  by  Mr.  James 
Emerson  in  connection  with  his  work  of  turbine-testing  when 
in  charge  of  the  testing-flume  at  Holyoke  (to  be  described  in  the 
next  paragraph).  The  engraving  in  Fig.  79  is  reproduced  from 
Mr.  Emerson's  book  "  Hydrodynamics "  (1881),  which  gives 
records  of  his  numerous  tests,  with  related  matter. 

97.  The  Holyoke  Testing-flume. — At  Holyoke,  Mass.,  where 
the  Connecticut  River  furnishes  a  large  water-power, '  falling 
some  60  feet,  the  Holyoke  Water-power  Co.  controls  the  water 
rights  and  leases  power  to  the  many  mill-operators  of  that 
city.  The  mill-owners  pay  a  certain  price  per  annum  per 
"  mill-power,"  which  in  that  locality  is  the  right  to  use  38  cub. 
ft.  of  water  per  second  under  a  head  of  20  ft.,  either  for  con- 
tinuous use  (a  24-hour  day)  or  for  a  definite  fraction  of  each 
day. 

The  fall  of  60  ft.  in  the  river  is  divided  into  three  parts  or 
steps,  two  intermediate  canals  having  been  constructed  at 
proper  levels,  in  such  a  way  that  the  tail-water  for  the  highest, 
or  next  highest,  series  of  mills  forms  the  head-water  of  the 
next  lower  series;  while  the  water  from  the  third,  and  lowest, 
series  is  discharged  into  the  lower  river.  In  order  that  the 
rate  at  which  any  mill  turbine  uses  water  at  any  stage  or  posi- 
tion of  its  gate  or  regulating  apparatus  may  become  known 
by  simply  observing  the  position  of  the  gate,  each  turbine, 


FIG.  80. 


H 


FIG.  81. 


152 


§  97.  TURBINE   TESTING.  153 

before  being  installed  in  the  mill  where  it  is  to  work,  is  tested 
at  the  "  testing- flume"  of  the  company  and  thus  becomes  a 
water-meter;  whose  indications,  when  the  motor  is  in  final 
place,  are  noted  from  day  to  day  by  an  inspector  of  the  com- 
pany. In  the  same  test  'its  power,  best  speed,  and  efficiency 
are  also  determined. 

The  testing-flume  occupies  the  lower  part  of  a  substantial 
building,  and  its  main  features  are  shown  in  vertical  section  in 
Fig.  80.  The  walls  of  the  wheel-pit  DD,  which  is  20  ft.  square, 
are  built  of  stone  masonry  and  lined  with  brick  laid  in  cement. 
The  water  is  admitted  to  it  from  the  head  canal  through  a 
trunk,  or  penstock,  and  vestibule,  which  are  not  shown  in 
the  figure.  Over  an  opening  in  the  floor  of  the  wheel-pit  the 
wheel  to  be  tested,  W,  is  set  in  place,  the  water  discharged 
from  it  finding  its  way  through  a  large  opening  into  the  tail- 
race  C,  35  ft.  long  and  20  ft.  wide,  and  finally  over  a  sharp- 
crested  weir,  at  A,  into  the  lower  canal.  The  whole  head  h 
available  for  testing  may  be  from  4  to  18  ft.  for  the  smaller 
wheels,  and  from  11  to  14  ft.  for  large  wheels,  up  to  300  H.P. 
The  measuring  capacity  of  the  weir,  which  may  be  used  to  its 
full  length,  20  ft.  (and  then  would  have  no  end-contractions), 
is  about  230  cub.  ft.  per  second.  The  head  h  becomes  known 
in  any  test  by  observations  on  the  water  level  in  two  glass 
tubes  communicating  with  the  respective  bodies  of  water  W 
and  C.  The  water  in  channel  C,  which  is  a  "  channel  of  approach" 
for  the  weir  A,  communicates  (at  a  point  some  distance  back 
of  the  weir)  by  a  lateral  pipe  with  the  interior  of  a  vessel  open 
to  the  air,  in  a  side  chamber.  Water  rises  in  this  vessel  and 
finally  remains  stationary  at  the  same  level  as  that  of  the 
surface  in  the  channel  of  approach.  A  hook  gauge  being  used  in 
connection  with  this  vessel,  observations  and  readings  are  taken 
from  which  the  value  of  h2,  or  "head  on  the  weir/'  maybe  com- 
puted :  for  use  in  the  proper  weir  formula  for  the  discharge,  Q. 

Fig.  80  shows  a  turbine,  in  position  for  testing,  with  a  ver- 
tical shaft — the  more  ordinary  case.  Upon  the  upper  end  of 
the  shaft  is  secured  a  cast-iron  pulley,  P,  to  the  rim  of  which 
the  Prony  brake  is  fitted  for  purposes  of  test. 


154  HYDRAULIC   MOTORS.  §  98^ 

98.  The  Prony  Brake  and  its  Use.— The  style  of  Prony  fric- 
tion-brake used  by  Mr.  Emerson  in  1880,  and  for  some  twenty 
years  afterwards  by  his  successors,  at  the  Holyoke  Testing-flume 
is  shown  in  Fig.  81.  Upon  the  rim  of  the  cast-iron  pulley,  B, 
keyed  upon  the  shaft  of  the  turbine  is  fitted  the  hollow  brass 
band  A  (shown  also  in  section  at  A),  the  friction  of  which  upon 
the  pulley  can  be  varied  by  the  tightening  or  loosening  of  the 
screw  at  M,  this  screw  being  turned  by  a  hand- wheel  N.  Both 
the  rim  of  the  pulley  and  the  friction  band  are  hollow,  water 
being  circulated  through  them  by  the  use  of  the  flexible  hose 
G  and  R.  The  pulley  revolves  clockwise,  as  seen  from  above, 
and  the  brake  in  its  tendency  to  revolve  with  the  pulley  exerts 
a  horizontal  pull  toward  the  left  (through  the  projecting  arm 
shown)  upon  the  upper  end  P  of  the  vertical  arm  of  the  bell- 
crank  lever  PFH  (fulcrum  at  F).  A  sufficient  weight  hung 
at  C  holds  the  bell -crank  in  equilibrium  and  a  pointer  playing 
along  a  scale  at  E  indicates  when  the  lever  is  in  its  median 
position.  At  D  is  attached  to  the  lever  a  vertical  rod  carrying 
at  its  lower  end  a  piston  fitting  loosely  in  a  fixed  vertical  cylin- 
der containing  oil  or  water.  This  is  called  a  "dash-pot,"  its 
object  being  to  prevent  sudden  motions  of  the  lever,  since  while 
the  resistance. of  the  oil  to  the  motion  of  the  piston  is  prac- 
tically nothing  for  a  slow  motion,  it  is  very  great  for  a  sudden 
movement.  In  this  way  oscillations  are  controlled.  At  V  is 
a  counter  from  which  the  number  of  revolutions  made  by  the 
wheel  in  a  given  time  becomes  known. 

The  procedure  of  testing  was  about  as  follows:  The  brake 
being,  carefully  balanced  and  adjusted  beforehand,  a  light 
weight  was  placed  on  the  scale-pan,  and  the  wheel  started  at 
full  gate;  sufficient  friction  was  then  produced  to  balance  the 
weight,  and  the  speed  of  wheel  noted.  "The  load  was  then 
increased  at  intervals  of  two  or  three  minutes,  by  25  Ibs.  at  a 
time,  until  the  speed  of  the  wheel  had  fallen  below  that  of 
maximum  efficiency  for  the  head;  the  weights  were  then  re- 
duced again  and  the  velocity  of  the  wheel  allowed  to  increase 
until  the  maximum  was  again  passed.  The  same  process  was 
then  repeated  within  a  smaller  range  of  speed  and  with  smaller 


§  99.  TURBINE   TESTING.  155 

variations  of  load,  until  the  speed  of  best  work  had  been  more 
exactly  ascertained,  and  the  performance  of  the  turbine  at 
maximum  efficiency,  under  full  head  and  at  full  gate,  had  been 
very  precisely  determined.  This  was  repeated  at  each  of  the 
part  gates,  usually  down  to  one  half  maximum  discharge."  * 

A  letter  from  Mr.  A.  F.  Sickman,  present  engineer  in  charge 
(1905)  of  the  Holyoke  Testing-flume,  states  that  up  to  April  11, 
1905,  1542  wheels  have  been  tested  in  the  flume;  and  adds: 
"We  use  the  Emerson  brass  brake  but  seldom  now,  having  a 
full  set  of  Prony  brakes — home-made,  cast-iron  pulleys  with 
wooden  jackets,  giving  very  satisfactory  work." 

99.  Test  of  the  Tremont  Turbine.— The  test  of  the  "Tremont 
Turbine/7  a  160-H.P.  turbine  of  the  radial  outward-flow  type 
(Fourneyron)  made  at  Lowell,  .Mass.,  in  1855  by  Mr.  J.  B. 
Franc  is,  was  an  event  of  special  interest  in  the  history  of  hydraulic 
science  and  has  Become  classic.  Though  the  test  is  by  no  means 
recent,  it  was  carried  out  so  thoroughly  as  to  make  its  details 
highly  instructive  to  the  student  of  hydraulics.  The  main 
features  of  this  test  will  now  be  presented  and  commented  on.f 

The  inner  and  outer  radii  of  the  turbine,  whose  shaft  was 
vertical  and  whose  general  arrangement  was  like  that  of  Fig.  45, 
were  3.37  and  4.14  ft.  respectively;  height  between  crowns, 
0.937  ft.  at  entrance  and  0.931  at  exit.  There  were  33  guide- 
blades  and  44  turbine-vanes.  As  to  angles,  a  =  28°,  /?  =  90°, 
d  =  22°  (see  Figs.  45,  47,  and  48).  The  areas  F0  and  Fn  were 
6.54  and  7.69  sq.  ft.,  respectively;  and  the  head,  h,  on  the 
wheel  about  13  ft.  (see  details,  later).  The  gate  was  a  thin 
cylinder,  movable  vertically,  between  the  guides  and  the  wheel. 
There  were  no  horizontal  partitions  dividing  up  the  wheel- 
channels;  in  fact,  no  special  device  for  preventing  the  loss  of 
head^  usually  arising  at  part  gate  with  this  kind  of  regulating 
apparatus. 

*  Quoted  from  Prof.  Thurston's  paper  on  the  "Systematip  Testing  of 
Turbine  Water-wheels  in  the  U.  S.,"  in  the  Transac.  Am.  Soc.  Mech.  Engrs., 
for  1887. 

f  Full  particulars  may  be  found  in  Mr.  Francis'  book,  "  Lowell  Hydraulic 
Experiments/'  New  York,  1880. 


156 


HYDRAULIC    MOTORS. 


99. 


TEST   OF   THE   TREMONT   TURBINE. 

(SELECTED  EXPERIMENTS.) 


1 

2 

3 

4 

5 

6 

7 

No. 

h 

n' 

Q 

R'v' 

f] 

of 
Exper. 

feet. 

revs, 
per  sec. 

cub.  ft. 
per  sec. 

ft.-lbs. 
per  sec. 

effic. 

H.P. 

FULL  GATE. 


1 

12.80 

0.00 

135.6 

0 

0.00 

2 

12.95 

0.45 

133.4 

73,160 

.68 

3 

12.97 

0.53 

133.7 

78,490 

.  .72 

4 

12.97 

0.60 

134.8 

82,110 

.75 

5 

12.94 

0.64 

135.1 

83,960 

.77 

6 

12.90 

0.85 

138  2 

88,210 

794 

160.3 

7 

12.90 

0.88 

139.0 

88,190 

.788 

8 

12.90 

0.90 

139.6 

88,076 

.784 

9 

12.85 

1.00 

141.9 

86,310 

.75 

10 

12.85 

1.06 

142.5 

83,970 

.73 

11 

12.80 

1.18 

144.8 

77,150 

.67 

12 

12.70 

1.31 

147.3 

66,840 

.57 

13 

12.65 

1.46 

152.3 

51,680 

.43 

14 

12.55 

1.60 

156.6 

33,350 

.27 

15 

12.54 

1.79 

162.3 

0 

0,00 

PART  GATE. 


16 

13.51 

0.00 

60.3 

0 

0.00 

17 

13.55 

0.46 

67.8 

24,460 

.43 

18 

13.48 

0.67 

71.8 

27,980 

.46 

50.9 

19 

13.39 

0.96 

76.6 

21,250 

.33 

20 

13.34 

1.25 

80.4 

0 

0.00 

§  99.  TEST  OF  TREMONT  TURBINE.  157 

A  large  and  strong  friction  brake  was  used  for  the  test, 
with  arcs  of  wood  rubbing  on  the  cast-iron  pulley  which  was 
keyed  to  the  turbine  shaft,  and  was  arranged  with  a  bell-crank 
lever  as  in  Fig.  81,  and  also  a  "  dash-pot."  The  various  lever- 
arms  concerned  were  of  such  values  that,  with  G  denoting  the 
necessary  weight  at  C  for  the  equilibrium  of  the  brake  in  any 
experiment,  the  corresponding  value  of  the  friction  at  the  rim 
of  the  pulley  was  Rf  =  3.9380  Ibs. 

The  rate  of  flow,  or  discharge,  Q  cub.  ft.  per  second,  was 
measured  by  two  weirs  at  the  end  of  the  tail-race,  somewhat 
as  in  Fig.  80,  use  being  made  of  the  " Francis  Formula"  for 
weirs  (see  p.  687,  M.  of  E.) ;  while  the  useful  power,  R'v'  (ft:-lbs. 
per  second  spent  on  friction  at  rim  of  pulley),  was  computed 
from  the  relation 

R'v'  =  (3.938G)  (2;r  2.75n'),  ft.-lbs.  per  sec.,  .  .  (1) 
in  which  G  is  the  weight  on  scale-pan  in  Ibs.  and  n'  the  number 
of  revs,  per  second  of  the  turbine  in  any  experiment  (steady 
operation).  The  radius  of  the  friction- pulley  was  2.75  ft.. 

The  annexed  table  gives  the  principal  data  and  results  of 
Mr.  Francis's  test  of  the  Tremont  Turbine,  arranged  in  the 
order  of  the  speed  of  wheel.  In  Experiments  Nos.  1  to  15 
(see  column  1)  the  cylindrical  gate  was  fully  open  ("full  gate  "), 
while  in  experiments  16  to  20  it  was  in  a  single  fixed  position 
leaving  open,  at  the  wheel-entrance,  about  one  quarter  of  the 
vertical  height  between  crowns;  in  other  words,  the  gate  was 
drawn  up  about  one  quarter  of  its  full  range  of  height.  In  this 
special  "  part-gate  "  position,  however,  the  quantity  of  water 
passing  per  second  was  much  greater  than  one  quarter  of  that 
passing  at  " full  gate";  as  is  seen  from  the  values  of  Q  in 
column  4.  For  example,  in  Exper.  18,  in  which  (for  this  posi- 
tion of  the  gate)  the  efficiency  was  a  maximum,  the  value  of 
Q  is  about  one  half  of  the  Q  used  in  Exper.  6  which  gives  the 
maximum  efficiency  at  full  gate.  It  would  be  said,  therefore, 
that  in  Exper.  18  the  wheel  was  working  at  about  "half  gate." 
The  heading  of  each  column  of  the  table  -shows  clearly  the 
nature  of  the  quantity  given  in  that  column  and  the  units  of 
measurement  involved  in  its  numerical  value. 


158  HYDRAULIC   MOTORS.  §  100. 

The  computations  relating  to  a  typical  experiment  will  now 
be  given,  Exper.  No.  6  being  selected.  In  this  experiment  the. 
weight  placed  on  the  scale-pan  was  1524  Ibs.  Hence,  when 
the  brake  was  tightened  sufficiently  so  that  the  wheel  raised 
the  weight  and  held  it  balanced,  the  friction  was  R'  =  3.938  X 
1524  =  6001  Ibs.  The  speed  of  the  wheel  having  adjusted 
itself  in  this  experiment  to  a  rate  of  n'  =  0.851  revs,  per  second, 
the  linear  velocity  of  pulley-rim  (its  radius  being  2.75  ft.)  was 
i/  =  27zm',  =2x  2.75X0.851,  =14.70  ft.  per  sec.  Hence  the 
useful  work  done  per  second  was  #V  =  6001X14.70  =  88,214 
ft. -Ibs.  per  sec. 

As  to  the  value  of  Q,  the  combined  length  of  the  two  sharp- 
edged  weirs  in  vertical  "thin  plate"  was  6=16.98  ft., -the 
number  of  end-contractions  was  n=4  (two  weirs),  and  the  head 
on  the  weir  h2=1.87  ft.  (velocity  of  approach  negligible). 
Hence,  from  the  formula 

Q  =  3.33(&-0.1nfa)ft2*,  cub.  ft.  per  sec.,      .     .     (2) 

which  is  the  same  as  eq.  (14)  of  p.  686,  M.  of  E.,  when  32.2 

is  put  for  g  (that  is,  for  the  foot  and  second  as  units),  we  have 

Q  =  3.33[16.98- 0.1  X4 X  1.87] X  (1.87)1  =  138.2  cub.  ft.  per  sec. 

The  difference  of  elevation  of  head-  and  tail-waters  was 
12.90  ft.,  so  that  Qrh  was  138.2x62.5x12.90,  =111,400  ft.-lbs. 
per  second;  and  consequently  the  efficiency,  >?,  =  (R'tf)  -5-  (Qrh), 
=  0.794;  or  79.4  per  cent. 

100.  Discussion  of  the  Test  of  Tremont  Turbine. — See  table 
on  p.  156.)  In  the  experiments  with  full  gate,  Nos.  1  to  14 
inclusive,  on  account  of  the  progressive  lessening  of  the  weight 
G  in  the  scale-pan  ^the  brake  friction  being  regulated  each 
time  to  correspond)  the  uniform  speed  to  which  the  wheel 
adjusts  itself  in  successive  experiments  increases  progressively 
from  the  zero  value,  or  state  of  rest,  of  Exper.  1,  when  the 
friction  was  so  great  as  to  prevent  any  motion,  up  to  a  maxi- 
mum rate  of  1.79  revs,  per  sec.,  attained  when  no  brake  fric- 
tion whatever  (ano  load")  was  present.  In  this  last  experi- 
ment, there  being  no  useful  work  done,  all  the  energy  of  the 


§  100.  '          TEST  OF  TREMONT  TURBINE.  159 

mill-site  is  wasted,  partly  in  axle  friction,  but  chiefly  in  fluid 
friction  (eddying  and  foaming  of  the  water;  finally,  heat) 
both  in  the  wheel-passages  and  also  in  the  tail-race,  where  the 
water  which  has  left  the  wheel  with  high  velocity  soon  has  its 
velocity  extinguished.  The  same  statement  is  true,  also,  for 
Exper.  No.  1,  .except  that  axle  friction  is  wanting.  In  both 
experiments  the  efficiency  is,  of  course,  zero. 

The  quantity  of  water  discharged  per  second,  Q,  is  seen  to 
increase  slowly  (after  Exper.  2)  from  133.4  to  162.3  cub.  ft. 
per  sec.,  though  not  differing  from  the  average  by  more  than 
ten  per  cent.  This  may  be  accounted  for  in  a  rude  way  as  an 
effect  of  " centrifugal  action "  (as  in  a  centrifugal  pump), 
since  the  Tremont  turbine  is  an  outward-flow  wheel.  The 
reverse  is  found  to  be  true  for  inward-flow  turbines,  notably 
the  Thomson  vortex  wheel  (see  §  90),  which  is  therefore  to  some 
extent  self -regulating  in  the  matter  of  speed;  since  a  less  dis- 
charge at  a  speed  higher  than  the  normal  diminishes  the  power 
and  hence  the  tendency  to  further  increase  of  speed. 

In  the  succession  of  experiments  Nos.  1  to  15  (all  at  full 
gate  and  under  practically  the  same  head  h)  the  efficiency  is 
seen  to  have  a  zero  value  both  at  beginning  and  end  of  this 
series,  and  to  reach  its  maximum  at  about  the  6th  experiment, 
in  which  the  speed  is  noted  as  being  about  one  half  that  at 
which  the  turbine  runs  when  entirely  "  unloaded  "  (Exper.  15). 
This  is  roughly  true  in  nearly  all  turbine  tests,  but  a  notable 
feature  of  considerable  practical  advantage  is  that  a  fairly 
wide  deviation  from  the  best  speed  affects  the  efficiency  but 
slightly.  For  instance,  a  variation  of  speed  by  25  per  cent, 
either  way  from  the  best  value  (of  0.85  revs,  per  sec.)  causes 
a  diminution  in  the  efficiency  of  only  about  four  per  cent. 

It  should  be  remembered,  also,  in  this  connection,  that  since 
the  water  used  per  sec.  (i.e.,  Q)  is  somewhat  different  at  differ- 
ent speeds  (at  full  gate),  the  speed  of  maximum  power  differs 
'slightly  from  that  of  maximum  efficiency. 

In  the  five  " part-gate''  experiments,  Nos.  16  to  20,  the 
gate  remains  fixed  in  a  definite  position  (about  one  quarter 
raised;  although  the  discharge  is  about  one  half  that  of  full 


160  HYDRAULIC   MOTORS.  §  101. 

gate)  through  all  these  five  runs.  The  head  is  practically  con- 
stant. At  first  the  wheel  is  prevented  from  turning.  The 
power  and  efficiency  are  then,  of  course,  zero;  but  Q  =  60.3 
cub.  ft.  per  sec.  As  the  turbine  is  permitted  to  revolve  under 
progressively  diminishing  friction  (#'),  the  speed  of  steady 
motion  becomes  greater,  reaching  its  maximum  (1.25  revs,  per 
sec.)  when  the  wheel  runs  "  unloaded/7  in  Exper.  20;  but  the 
power,  or  product,  R'vf ',  reaches  a  maximum  and  then  diminishes. 
The  same  is  true  of  the  efficiency,  whose  maximum  (in  Exper. 
18)  is  seen  to  be  about  46  per  cent.,  only.  This  forms  a  strik- 
ing instance  of  the  disadvantage  and  wastefulness  of  a  cylindrical 
gate,  unaccompanied  by  other  mitigating  features,  when  in 
use  at  part  gate.  This  defect,  however,  may  be  largely  reme- 
died by  the  use  of  horizontal  partitions  in  the  wheel-channels, 
as  in  Fig.  46,  or  by  employing  curved  upper  crowns,  as  in  the 
American  "inward  and  downward  "  turbines. 

101.  Tremont  Turbine  Test.  Graphic  Representation. — 
Taking  the  angular  speed  revs,  per  sec.  as  an  abscissa,  and  the 
efficiency  as  an  ordinate,  points  on  paper  may  be  plotted  and 
the  curve  thus  formed  called  an  "  efficiency  curve/'  showing 
the  variation  of  the  efficiency  with  the  speed  of  the  turbine. 
Such  a  curve  is  shown  as  OAaB  in  Fig.  82,  having  been  plotted 
from  the  fifteen  full-gate  experiments  of  the  table  on  p.  156, 
The  point  of  maximum  efficiency  occurs  at  a,  and  the  scale  of 
efficiency  is  marked  on  the  vertical  axis  at  the  left.  Similarly,, 
if  the  values  of  Q  are  used  as  ordinates,  with  the  speeds  as 
abscissae,  the  curve  CE  results,  showing  very  plainly  the  gradual 
increase  of  Q,  with  the  speed,  after  the  second  and  third  ex- 
periments. The  scale  for  Q  is  given  along  the  right-hand  edge 
of  the  diagram.  Similarly,  the  smaller  curves  GF  and  \M 
show  the  variation  with  speed  of  the  efficiency  and  discharge, 
respectively,  for  the  five  part-gate  experiments.  In  all  the 
curves  the  point  corresponding  to  each  experiment  is  shown 
by  a  small  circle. 

(Details  of  many  other  tests  may  be  found  in  Mr.  Bodmer's1 
book,  in  Mr.  Emerson's  Hydrodynamics,  and  in  technical 
journals.) 


CM         Q         CO         <O         ^        CM 


^     o 


II 


161 


162  HYDRAULIC    MOTORS.  §  102 

• 

102.  Regulating  "  Gates  "  of  a  Turbine. — When  a  variable 
power  is  demanded  of  a  turbine,  as  when  in  a  factory  the  num- 
ber of  machines  operated  is  not  constant,  or  when  the  amount 
of  electric  current  generated  in  a  dynamo  run  by  the  turbine 
is  required  to  be  variable  to  suit  the  varying  demands  of  street- 
railway  work  or  electric  lighting,  the  average  position  of  the 
turbine  gate  is  not  that  of  "full  gate."  Since  the  speed  of  the 
turbine  should  be  fairly  constant,  especially  for  electric  work 
(and  this  has  to  do  with  the  question  of  governors  treated  in 
§  103),  the  required  variation  in  power  must  be  provided  by 
varying  the  amount  of  water  used  per  second,  i.e.,  Q;  and  this 
requires  movement  of  the  gate  or  regulating  apparatus.  It  is 
therefore  of  importance,  where  economy  in  the  use  of  water  is 
necessary,  that  a  turbine  should  have  a  fairly  high  efficiency 
at  "part  gate." 

At  the  outset  the  statement  should  be  emphasized  that 
perhaps  the  most  wasteful  device  for  varying  the  discharge  of 
water  is  the  "throttling  "  of  the  flow  by  the  use  of  a  gate  in 
the  penstock  or  supply-pipe,  or  in  the  draft-tube  (see  §  51  in 
this  connection) ;  or  by  the  use  of  a  cylindrical  gate  encircling 
the  lower  end  of  a  draft-tube;  since  these  either  induce  losses 
of  head  due  to  sudden  enlargement  of  waterway,  or  bring  about 
impact  of  the  water  at  the  turbine  entrance,  where  for  the  usual 
speed  of  wheel  the  value  of  the  angle  /?  is  only  suited  to  a  fixed 
value  of  the  velocity  w\. 

The  plain  cylindrical  gate  moving  axially  is  open  to  similar 
objections,  unless,  as  already  stated,  the  turbine  channels  are 
provided  with  partitions  or  their  equivalents,  or  have  an  upper 
crown  which  curves  downward. 

Perhaps  the  most  perfect  "gate,"  from  a  theoretical  point  of 
view,  for  a  radial  turbine  is  the  device  of  Nagel  and  Kaemp, 
in  which  not  only  are  the  "roofs  "  of  the  guide-passages  movable, 
but  also  the  corresponding  crown  of  the  turbine.  The  crown 
being  always  placed  even  with  the  roof,  sudden  enlargement 
at  the  turbine  entrance  is  prevented  in  all  positions  of  the  regu- 
lating apparatus.  The  turbine  thus-  becomes  one  of  variable 


§   103.  REGULATION    OF   TURBINES.  163 

height,  e,  between  crowns.  This  design  is;  however,  expensive 
and  attended  with  practical  difficulties. 

The  three  kinds  of  gate  often  used  with  American  "inward 
and  downward "  turbines  (viz.,  the  cylinder,  register,  and 
wicket  gate)  have  been  already  mentioned  in  §  86.  See  also 
§79. 

The  regulation  of  the  Jonval  or  parallel-flow  (" axial") 
type  of  turbine  is  usually  accomplished  by  sliding  plates  or 
swinging  flaps  for  closing  of  the  guide-passages.  The  entire 
closing  of  a  number  of  the  guide-passages,  instead  of  the  partial 
closing  of  all  of  them,  is  found  to  conduce  to  a  higher  efficiency; 
since  in  the  former  case  the  value  of  the  absolute  velocity  (wi) 
at  entrance  of  the  turbine  remains  the  same  as  when  all  the 
guide-passages  are  in  use.  (See  Bodmer  for  many  further 
details;  also  Buchetti,  and  Mueller.)  The  "  Duplex "  Jonval 
wheel,  made  by  R.  D.  Wood  and  Co.  of  Philadelphia,  has  already 
been  referred  to  in  §  86. 

In  this  connection  attention  should  be  called  to  Mr.  Thurso's 
valuable  article,  already  mentioned  in  §  88. 

103.  "  Mechanical  "  Governors  for  Turbines. — The  power  to 
move  the  turbine  gates  is  usually  furnished  by  the  turbine 
itself;  but,  more  frequently,  in  large  modern  plants,  by  a 
hydraulic  "relay"  motor,  or  hydraulic  piston  and  cylinder 
actuated  by  water  or  oil;  pressure- water  from  the  penstock,  or 
oil  from  a  pressure-tank  (compressed  air  above  the  oil).  The 
"  governor  "  proper  consists  of  revolving  centrifugal  balls  and 
their  connections  whose  change  of  relative  position,  brought 
about  by  a  slight  change  in  the  speed  of  the  turbine  (with 
which  the  governor  is  in  gear),  moves  the  proper  valves  or  other 
parts  necessary  to  bring  into  play  the  motor  or  mechanism 
which  moves  the  turbine  gates.  Electric  governors  have  been 
used,  but  not  extensively. 

A  mechanism  which  in  its  general  form  has  been  long  in 
use  in  cases  where  the  turbine  itself  furnished  the  power  directly 
for  moving  the  gate,  and  furnishing  an  instance  of  a  "mechan- 
ical governor,"  is  illustrated  in  the  King  water-wheel  governor, 
shown  in  Fig.  83.  The  turning  of  the  horizontal  shaft  7,  caused 


164 


HYDRAULIC    MOTORS. 


§  103. 


B 


B 


by  the  rotation  of  the  spur-wheel  1,  moves  the  turbine  gate. 
The  vertical  shaft,  S,  carrying  the  centrifugal  balls  (B,  B) 
rotates  at  a  speed  proportional  to  that  of  the  turbine,  being  in 

gear  with  the  latter;  and  also  causes 
the  continuous  rotation,  in  one  direc- 
tion, of  the  wheel  5,  connected  by  a 
crank  and  connecting-rod  with  arm, 
or  crank,  4.  There  is  thus  brought 
about  a  continual  to-and-fro  horizon- 
tal motion  of  the  upper  end  of  arm  4, 
to  which  are  pivoted  two  "pawls," 
2  and  3,  either  of  which,  if  hanging 
low  enough  to  do  so,  would  by  a  suc- 
cession of  direct  thrusts  against  the 
cogs  turn  the  wheel  1,  and  thus  either 
open  or  close  the  turbine  gate,  accord- 
ing to  which  pawl  might  be  in  action. 
When  the  speed  of  the  turbine  is 
normal  neither  pawl  can  turn  the 
wheel  1,  since  in  that  case  its  ex- 
tremity is  held  out  of  contact  with 
the  cogs  of  1  by  a  projecting  "peg  " 
which  slides  along  the  edge  of  the  thin  disc  6.  At  normal 
speed  of  turbine  the  disc  6  is  stationary  and  in  its  median  posi- 
tion; but  when  that  speed  changes  and  the  balls  consequently 
change  their  distances  from  the  axis  of  the  vertical  shaft,  the 
vertical  spindle  8  is  moved  either  up  or  down  and  rotates 
disc  6  sufficiently  to  bring  one  or  the  other  of  two  depressions 
(in  the  edge  of  the  disc)  under  the  "peg  "  of  one  of  the  pawls, 
thus  allowing  the  pawl  to  drop  and  actuate  wheel  1,  which 
then  moves  the  gate  in  the  p'roper  direction. 

The  "Snow  Water-wheel  Governor"  has  been  extensively 
used  both  in  England  and  America,  using  practically  the 
same  design  of  pawls,  etc.,  as  in  the  King  governor,  and  is 
shown  in  Fig.  84.  The  turbine  gate  is  moved  by  the  turn- 
ing of  the  vertical  shaft  PA,  which  can  also,  on  occasion, 
be  actuated  by  the  hand- wheel  at  upper  end.  The  two  lower 


FIG.  83. 


SNOW'S 
WATER-WHEEL   GOVERNOR 


FIG.  84 


165 


166  HYDRAULIC    MOTORS.  §   104. 

horizontal  shafts  at  G  turn  continuously,  being  in  gear  with 
the  turbine,  but  the  third  one,  B,  turns  only,  and  in  the  proper 
direction,  when  the  speed  of  the  turbine  changes  slightly  from 
the  normal,  and  moves  the  turbine  gate  by  means  of  the 
bevel-gear  at  B. 

104.  Hydraulic  Governors  for  Turbines. — The  foregoing  are 
called  "  mechanical  governors/7  the  power  for  moving  the  gate 
being  furnished  directly  by  the  turbine  itself.  A  "hydraulic 
governor "  made  by  a  prominent  American  firm,  the  Lom- 
bard Governor  Co.  of  Ashland,  Mass.,  and  called  "Type  N" 
(among  their  various  designs),  is  shown  in  Fig.  85.  The  ver- 
tical hydraulic  cylinder,  with  piston  (" main  piston"),  etc., 
constituting  the  "  relay  motor,"  occupies  the  lower  half  of  the 
mechanism  in  the  figure.  To  the  cross-bar  secured  to  the 
upper  end  of  the  (main)  piston-rod  are  attached  two  vertical 
racks  by  whose  motion  the  horizontal  shaft  (seen  projecting  out 
at  the  right)  is  made  to  turn  and  actuate  the  turbine  gate. 
This  shaft  can  also  be  rotated,  if  necessary,  by  the  large  hand- 
wheel  seen  in  front.  The  small  pulley  near  the  top  (on  left) 
is  belted  to  another,  actuated  by  the  turbine  shaft,  and  con- 
tinual rotary  motion  of  the  centrifugal  balls  results.  These  balls 
when  rotating  at  normal  speed  stand  out  considerably  from 
the  axis  of  rotation.  The  mechanism  is  such  that  if  the  balls 
spread  out  under  the  action  of  an  increase  of  speed,  they  depress 
the  top  plate  into  which  the  flat  springs  supporting  them 
are  inserted;  and  vice  versa.  This  top  plate  is  attached  to  a 
rod  which  passes  down  through  the  hollow  vertical  shaft  carry- 
ing the  balls,  and  terminates  in  a  small  "  primary  valve,"  a 
slight  motion  of  which  from  its  normal  position  causes  ad- 
mission of  oil  (under  pressure)  to  actuate  the  hydraulic  plungers 
of  a  "  relay-valve  "  device,  whose  motion  causes  movement  of 
the  main  valve.  The  office  of  the  main  valve  is  to  admit  oil 
from  a  pressure-tank  to  one  side,  or  the  other,  of  the  main 
piston  whose  motion,  through  the  vertical  racks  and  gear-wheel, 
causes  motion  of  the  turbine  gate.  The  other  side  of  the  main 
piston  is  at  the  same  time  put  into  communication  with  the 
11  vacuum-tank." 


FIG.  85.     The   Lombard  "N"   Governor. 


[Note  1914.  The  present  form  of  this  governor  contains  improvements  ; 
among  other  things,  a  worm-gear  hand  control,  and  an  extension  piston 
head  that  partly  closes  the  ports  at  the  ends  of  the  stroke  for  the  purpose 
of  preventing  serious  inertia  effects  when  called  into  fastest  action.] 


§  104a.  TURBINE  REGULATION.  167 

A  pressure-tank  (not  shown  in  the  figure)  contains  com- 
pressed air  and  oil  under  about  200  Ibs.  per  sq.  in.  pressure  and 
supplies  oil  for  the  main,  and  relay,  hydraulic  cylinders.  Pumps 
run  by  the  turbine  itself  pump  the  oil  back  into  the  pressure- 
tank  from  the  vacuum-tank  as  occasion  requires.  One  complete^ 
stroke  of  the  main  piston  entirely  opens  or  closes  the  water- 
wheel  gates;  consequently  any  motion  of  this  piston  less  than 
a  complete  stroke  causes  a  proportionally  smaller  motion  of 
the  gates. 

The  Allis-Chalmers  Co.  of  Milwaukee,  Wis.,  manufacture 
the  hydraulic-governor  designs  of  the  Swiss  firm  Escher,  Wyss, 
and  Co. 

A  description  of  the  "  Replogle  Differential  Relay  "  gov- 
ernor, made  by  the  Replogle  Governor  Works  at  Akron,  Ohio, 
may  be  found  in  the  Engineering  News  of  Nov.  13,  1902,  p. 
409.  This  governor  has  a  heavy  " inertia  wheel"  to  supple- 
ment the  action  of  the  ordinary  fly-balls  when  very  prompt 
motion  of  the  turbine  gate  is  called  for. 

io4a.  Fly-wheels. — If  a  fly-wheel  is  placed  upon  the  shaft 
of  a  turbine,  the  inertia  of  the  mass  so  added  tends  to  retard 
a  change  of  speed  on  the  part  of  the  turbine  when  the  "load  " 
changes,  thus  giving  the  governor  and  its  accessories  more 
time  to  act,  and  enabling  the  speed  to  be  kept  within  a  smaller 
range  of  variation.  The  revolving  part  of  an  electric  generator 
is  sometimes  made  to  serve  the  purpose  of  a  fly-wheel,  as  oc- 
curred with  the  turbines  in  Power-house  No.  1  of  the  Niagara 
Falls  Power  Co.,  no  other  fly-wheel  being  found  necessary. 

Mr.  Thurso  mentions  the  case  of  a  1000- H. P.  turbine  at 
Jajce,  Bosnia,  (see  reference  in  §  88,)  as  using  a  hydraulic  gov- 
ernor which  keeps  the  speed  within  1^  per  cent,  of  the  normal. 
A  small  fly-wheel  is  employed  to  assist  &e  governor. 

A  valuable  article  on  "Speed  Regulation  of  High  Head 
Water  Wheels,"  by  Mr.  H.  E.  Warren,  Superintendent  for  the 
Lombard  Governor  Co.,  will  be  found  in  vol.  xx,  No.  2,  June 
1907,  of  the  "  Technology  Quarterly, "  published  at  the  Mass. 
Inst.  of  Technology,  Boston,  Mass.  In  this  article  a  discussion 
is  given  of  the  fly-wheel  effects  of  the  revolving  parts  when 
changes  of  "  load  "  occur. 


CHAPTER  VII. 
CENTRIFUGAL  AND  "  TURBINE  "  PUMPS. 

105.  Turbine  as  Centrifugal  Pump. — Let  us  suppose  that 
we  have  a  radial  outward-flow  turbine  in  steady  operation,  as 
in  Fig.  45  on  page  92,  and  that  suddenly  the  depth  of  the  tail- 
water  is  largely  increased  so  that  its  surface  T  is  at  a  higher 
elevation  than  that,  H,  of  the  "  head- water  "  or  supply  reser- 
voir. To  keep  up  the  same  flow  of  water  as  before,  radially 
outward  through  the  turbine  passages,  will  necessitate  the 
application,  to  the  turbine,  of  working  forces  from  some  external 
source  of  power,  such  as  a  steam-engine.  That  is,  instead  of 
providing  a  resistance  Rf  Ibs.  at  the  periphery  of  the  upper 
pulley  M  on  the  turbine  shaft  to  prevent  acceleration,  we  must 
now  furnish  a  working  force  P  Ibs.  (pointing  toward  the  right 
on  the  near  side  of  the  pulley  M)  to  prevent  retardation.  The 
work  done  by  P  each  second  is  Pv  ft. -Ibs.  per  sec.  and  is  em- 
ployed in  maintaining  the  steady  flow.  Since  water  is  now 
being  raised  from  a  lower  to  a  higher  level,  the  turbine  has 
become  a  pump;  called  a  "centrifugal  pump." 

In  actual  centrifugal  pumps  there  are  ordinarily  no  guide- 
blades  at  G  (Fig.  45)  inside  the  wheel,  but  of  late  years  (since 
[900,  about)  many  such  pumps  have  been  built  with  guide- 
passages  outside  the  wheel  (or  "  impeller,"  as  it  is  called)  with 
gradually  enlarging  passageways  constituting  a  "diffuser/' 
to  diminish  losses  of  head  at  that  point;  with  consequent 
Improvement  in  efficiency.  To  this  more  recent  variety  of 
centrifugal  pump  the  name  " turbine  pump"  is  now  frequently 
applied  (1905). 

In  a  centrifugal  pump  the  action  of  the  water  on  the  "im- 

168 


§  106.  CENTRIFUGAL    PUMPS.  169 

peller"  is  equivalent  to  a  resisting  couple,  instead  ~>f  a  working 
couple,  and  the  moment  of  the  working  force  P  about  the  axis 
of  the  shaft  is  numerically  equal  to  that  of  this  couple  (aug- 
mented by  moment  of  axle  friction);  the  rotation  being  uni- 
form and  the  flow  "  steady." 

From  the  figure  (45),  the  vanes  of  the  turbine  (now  pump) 
being  curved  backwards  as  regards  the  direction  of  rotation, 
it  is  seen  that  these  vanes  tend  to  crowd  the  water  radially 
outward;  but  even  if  the  vanes  were  straight  and  were  radial, 
the  same  general  effect  would  be  produced  if  the  speed  were 
sufficient;  since,  from  its  " inertia/'  a  particle  of  water  tends 
to  persist  in  a  straight-line  motion  and  thus  incidentally  to 
increase  its  distance  from  the  axis  of  the  wheel.  In  a  rough 
general  way  this  outward  flow  of  the  water  between  radial 
vanes  is  sometimes  said  to  be  due  to  "  centrifugal  force,"  and 
rude  methods  of  analysis  have  been  based .  on  this  idea.  It 
is  better,  however,  to  avoid  these  imperfect  notions  of  "cen- 
trifugal force"  and  to  use  the  relations  that  have  been  proved 
to  apply  to  the  steady  flow  of  water  in  uniformly  rotating 
channels  and  pipes;  as  already  established  in  §§  31-42a  (see 
particularly  §§  35a  and  42a).  These  relations  were,  of  course, 
based  on  the  fundamental  laws  of  mechanics  as  applied  to  a 
material  point. 

106.  Notation  for  Centrifugal  Pump.— The  number  of  vanes 
used  in  the  majority  of  centrifugal  pumps  is  so  small  (four  to 
ten,  perhaps)  that  the  guidance  of  the  water  is  far  from  perfect 
and  consequently  the  theory  now  to  be  presented  must  be 
considered  as  giving  results  that  are  only  roughly  approxi- 
mate; especially  as  the  frictional  conditions  in  these  pumps 
are  only  imperfectly  understood.  Certain  general  indications, 
however,  are  clearly  brought  out  by  theory. 

The  pump  to  be  considered  is  one  with  a  horizontal  shaft, 
and  is  placed  above  the  source  of  supply,  a  suction-tube  being 
therefore  necessary.  Fig.  87  gives  a  vertical  section  through 
the  axis  of  the  shaft;  the  shaft  and  the  crowns  or  side  plates 
of  the  "impeller,"  or  revolving  part,  being  shown  in  solid 
black  shading.  Steady  flow  of  the  water,  with  full  pipes  and 


FIG.  86. 


170 


§  106. 


CENTRIFUGAL   PUMPS. 


171 


passageways,  and  uniform  angular  velocity  oj  of  the  impeller, 
are  postulated.  Fig.  86  gives  a  vertical  section,  at  right  angles 
to  the  shaft,  showing  the  (six)  impeller-blades  or  vanes,  such 
as  A.  .N,  the  supply-reservoir  T,  and  receiving-reservoir  H; 
also  suction-tube  (or  supply-pipe)  DD,  conducting  the  water 
from  T  to  the  central  space,  S,  of  the  impeller;  and  the  delivery- 
pipe  EJ.  The  casing,  XYZ,  within  which  the  impeller  rotates 
is  of  the  scroll  or  "volute"  form  so  commonly  used;  and  may 


IMPELLER 
.« 


FIG.  87 


be  looked  upon  as  a  single  external  guide-blade,  the  average 
radial  width  of  the  volute  space  increasing  from  E  toward  X, 
G,  and  K,  to  provide  for  the  increasing  number  of  water  fila- 
ments issuing  from  the  outer  edges  of  the  impeller-vanes;  hence 
the  velocities  of  these  filaments  are  about  equal.  All  of  these 
filaments  have  to  pass  through  the  horizontal  section  at  E  at 
the  base  of  the  delivery-pipe  J".  Upon  the  shaft  is  supposed 
to  be  secured  a  gear-wheel,  Wt  at  whose  periphery  ("  pitch- 
circle  ")  a  constant  tangential  pressure,  or  working  force, 
P  Ibs.,  is  assumed  to  be  acting;  furnished  by  a  motor  of  some 
kind  (a  steam-engine,  say),  and  of  suitable  amount  to  main- 
tain uniform  motion  of  the  pump  and  steady  flow  of  the  water. 


172  HYDRAULIC   MOTORS.  §  107. 

The  linear  velocity  of  the  point  of  application  of  P  being  de- 
noted by  v  (  =cor,  if  r  is  the  corresponding  radius),  the  power 
exerted  by  P  is  Pv  ft.-lbs.  per  sec.  At  the  entrance,  1,  of  the 
impeller  channels  the  absolute  velocity  w\  of  the  water  is  sup- 
posed to  be  radial,  since  there  are  no  internal  guides.  Tne 
tangent  to  the  impeller-blade  at  that  point  is  supposed  to  be 
placed  at  such  an  angle  /?  with  1.  .t,  the  tangent  to  wheel-rim, 
or  circle  of  rotation,  at  1,  so  as  to  avoid  impact.  That  is,  the 
former  tangent  should  follow  the  direction  of  the  relative 
velocity  c\  at  point  1.  The  linear  velocity  of  wheel-rim  at  1 
is  vi  =  cori,  and  the  width  between  crown-plates  is  ei  (see  Fig. 
87).  Similarly,  vn,  en,  and  rn  refer  to  the  exit  wheel-rim,  or  N. 

The  absolute  path  of  a  particle  of  water  from  entrance  to 
exit  of  wheel  is  shown  by  the  dotted  line  1.  .N,  the  vane 
along  which  it  moves  having  passed  from  position  1.  .F  to 
position  A.  .N.  The  absolute  velocity  wn  of  the  particle 
at  N  is  the  diagonal  of  the  parallelogram  on  the  wheel-rim 
velocity  at  N,  viz.,  vn,  and  the  relative  velocity  cn  which  is 
tangent  to  the  vane  curve  at  N  and  makes  some  angle  d  with 
the  wheel-rim  tangent  Nt.  The  angle  between  wn  and  wheel- 
rim  tangent  is  /*,  so  that,  its  component  tangent  to  the  wheel- 
rim  is  un  =  wn  cos  fj.}  called  the  "velocity  of  whirl,"  and  its  radial 
component  is  Vn  =  rWn  sin  /*,  called  the  velocity  of  flow. 

Evidently  at  the  entrance,  1,  the  velocity  of  whirl  is  zero 
and  the  velocity  of  flow  is  Vi  =  wi}  itself. 

Figs.  87a,  87b,  and  87c  show  a  section  through  the  shaft,  a 
section  transverse  to  the  shaft,  and  a  perspective  of  the  im- 
peller, respectively,  of  the  centrifugal  pump  made  by  the 
De  Laval  Steam  Turbine  Co.,  Trenton,  N.  J.,  and  put  on  the 
market  in  1902;  slightly  modified  since,  but  still  (1914)  of  the 
same  general  design.  (The  "  enclosed"  type,  §  114.) 

107.  Form  of  Loss  of  Head  to  be  Considered. — In  the  theory 
now  to  be  given  the  only  loss  of  head  (and  corresponding  waste 
of  power)  that  will  be  considered  will  be  that  due  to  the  more 
or  less  abrupt  change  of  absolute  velocity  occurring  in  the 
water  just  after  exit  from  the  impeller  passages.  In  the  pumps 
of  older  design  in  which  the  water  at  exit  is  discharged  into  a 
body  of  water  having  a  much  smaller  absolute  velocity  this  loss 


FIG.  8ya. 


FIG.  Syb. 


173 


174  HYDRAULIC   MOTORS.  §   108. 

of  head  is  perhaps  greater  than  that  due  to  any  other  cause. 

It  may  be  written  in  the  form  ^7T~)  m  which  the  value  of  the 

Z9 

coefficient  £  would  be  given  by  Borda's  formula  (p.  721,  M.  of 
E.).  If  the  velocity  finally  assumed  by  the  water  in  the  volute 
space  is  only  one  fifth  of  wn,  or  smaller,  the  value  of  £  is  prac- 
tically unity  or  1.0.  If,  however,  the  change  of  absolute 
velocity  at  exit  is  made  gradual  by  gently  flaring  passage- 
ways between  fixed  guide-blades,  the  value  of  £  may  be  as  low 
as  0.2  or  0.3  (if  we  may  judge  from  experiments  made  on  the 
loss  of  head  occurring  in  the  down-stream  diverging  portion 
of  a  Venturi  meter;  see  p.  726,  M.  of  E.).  But  to  offset  the 
fact  that  the  losses  of  head  occurring  in  the  impeller  channels 
themselves  will  be  ignored  in  the  theory  now  to  be  developed, 
it  would  probably  be  advisable  to  take  no  lower  value  than 
0.5  to  0.6  for  £,  even  in  the  case  of  a  " turbine  pump"  (that 
is,  one  provided  with  external  guide-blades);  while  for  the 
ordinary  pump  with  the  usual  abrupt  change  of  section  from 
impeller  to  volute  space  £  may  range  as  high  as  1.5  (especially 
with  high  he  ds;  over  20  ft.)  in  order  to  include  losses  *  in  impel- 
ler channels  with  the  loss  after  exit  due  to  sudden  enlargement. 

The  neglect  of  losses  of  head  in  both  suction-pipe  and 
delivery-pipe  implies  that  they  are  so  wide  and  short  that 
the  skin  friction  therein  is  negligible.  (In  this  connection,  see 
§§  115,  etc.) 

108.  Theory  of  the  Centrifugal  Pump.  Speed  of  "  Impend- 
ing Delivery." — If  the  centrifugal  pump  itself  and  both  pipes 
have  been  originally  filled  with  water,  a  foot-valve  being  pro- 
vided at  the  base  of  the  suction-pipe  to  prevent  a  backward 
flow  before  the  pump  is  started,  the  question  arises  as  to  how 
great  the  speed  of  rotation  must  be  before  any  upward  flow 
at  all  is  brought  about.  In  other  words,  what  must  be  the 
velocity,  vn,  of  the  tips  of  the  impeller-blades,  such  that  the 
only  effect  is  to  prevent  any  downward  flow  on  the  part  of 
the  water  in  the  delivery-pipe  and  upper  reservoir?  When 
this  state  of  equilibrium  occurs  the  water  in  both  suction-  and 

*  Such  losses  may  be  considerable  if  the  interior  surf  aces  are  those  of  rough 
castings. 


FIG.  870.     The  DeL,aval  Impeller. 


§  108.  CENTRIFUGAL   PUMPS.  175 

delivery-pipes  will  be  at  rest,  and  that  in  the  impeller  passages 
will  rotate  with  the  impeller  without  travelling  either  to  or 
from  the  axis.  Hence  the  fluid  pressure,  pn,  between  the  re- 
volving water  and  the  stationary  water  in  the  upper  pipe  is 
the  hydrostatic  pressure  due  to  the  depth  hn  of  point  N  below 
surface  H,  plus  atmospheric  pressure  (let  b  denote  the  water- 
barometer  height);  that  is,  pw=(An+&)r;  also;  the  fluid  pres- 
sure pi  at  point  1  is  that  due  to  the  depth  of  point  1  below 
an  imaginary  water  surface  34  ft.  (i.e.,  b  ft.)  above  T7,  or 
pi  =  (b—h\)Tt  where  hi  is  the  height  of  point  1  above  surface 
T  of  lower  reservoir.  (In  Fig.  86  the  pump  is  above  the  supply- 
reservoir  T;  if  it  were  at  a  lower  level,  pi  would  be  =  (b+hi)?, 
but  the  final  result  would  be  the  same.) 

In  this  case  we  may  consider  the  water  in  the  pump-channels 
to  have  a  steady  flow  outwards  from  1  to  N  with  relative 
velocities  (c\  and  cn)  =  zero,  and  apply  Bernoulli's  Theorem 
for  a  rotating  channel,  etc.,  i.e.,  eq.  (16)  of  §  42a;  in  which 
both  ci  and  cn  will  be  zero,  and  pi  and  pn  will  have  the  values 
just  given;  while  the  loss  of  head  In!'  will  be  zero  since  there 
is  no  flow;  whence  we  have 

o  o 

n~Vl  .....     (1) 


'Solving,  we  have,  after  noting  that  hi+hn=h,  and  that 


....     (2) 

as  the  value  for  the  linear  velocity  of  the  tip  of  the  impeller- 
blades  necessary  to  keep  the  water  from  flowing  back;  or  it 
may  be  called  the  "  velocity  for  impending  delivery,"  since,  if 
the  speed  is  increased  beyond  this,  a  flow  will  take  place  up 
the  delivery-pipe. 

For  example,  if  in  Fig.  86  r\  is  taken  as  one-third  of  rn,  and 
the  minute  and  foot  be  used  as  units,  we  have  (very  nearly) 

vn/  =  [500v//z,  (iiffL)]  feet  per  minute.    ...     (3) 
With  a  very  small  r\  (call  it  zero)  we  derive  481  instead  of 


176  HYDRAULIC   MOTORS.  §  109. 

the  500.  Experiment  shows  that  frictional  conditions  and 
also  the  shape  of  the  blade  have  an  influence  on  the  value 
of  vnf  for  impending  delivery.  Results  quoted  on  p.  98  of 
Engineering  News  for  August  1900  are  as  follows:  Instead  of 
the  500  in  eq.  (3)  above,  the  following  numbers  were  found,  in 
the  case  of  pumps  24  in.  in  diameter  with  n  =  about  one-half  of  rn : 

For  blades  curved  about  as  Fig.  86  (d  =  27°) 610 

".     "    "     "  (fl  =18°) 780 

Straight  radial  blades 480 

Straight  blades  leaning  backward  about  45° 554 

Curved  blade  somewhat  like  that  in  Fig.  86  and  with 
its  chord  in  same  position,  but  concave  on  the 

advancing  side 394 

Theoretically,  in  such  a  case,  since  no  water  is  pumped,  no- 
power  (Pv)  is  required  to  maintain  the  rotation  of  the  pump; 
that  is,  if  once  started  it  should  continue  the  motion  indefi- 
nitely; but  practically,  on  account  of  the  friction  between 
the  rotating  water  and  the  stationary  water  in  the  pipes  and 
between  the  discs  or  crowns  and  the  surrounding  water,  to- 
gether with  axle  friction  some  little  power  is  necessary  to 
keep  up  the  motion.  After  pumping  is  once  started  the  velocity 
of  the  tips  may  sometimes  be  allowed  to  sink  below  the  value 
of  eq.  (3)  if  the  pump  contains  provision  for  a  gradual  enlarge- 
ment of  section  in  the  casing  at  exit  from  the  wheel. 

109.  Theory  of  the  Centrifugal  Pump,  with  Friction.  Best 
Velocity.  Maximum  Efficiency. — Returning  to  Figs.  86  and 
87  and  assuming  a  steady  flow  of  water,  all  passages  full,  and 
uniform  rotation  of  pump  with  angular  velocity  a>,  with  other 
notation  of  §  106;  also  Q  denoting  the  rate  of  discharge,  or 
cub.  ft.  per  sec.  of  water  pumped.  At  first  all  the  quantities 
concerned,  except  h,  r\,  rn,  and  d,  will  be  considered  variable. 
Later,  special  conditions  will  be  imposed. 

Applying  the  equation  for  power  based  on  "  angular  momen- 
tum," etc.  (eq.  (10)  of  §  34);  (see  also  §  35)  (work  per  second 
done  on  equivalent  couple),  and  remembering  that  in  this, 
case  Ui  is  zero  and  that  the  couple  representing  the  action  of 


§  109.  CENTRIFUGAL    PUMPS.  177 

the  water  on  the  wheel  is  a  resisting  instead  of  a  motive  couple, 
we  have,  neglecting  axle  friction, 

QT 

Pv=  —  vnun  (ft.-lbs.  per  sec.)  power     .     .     .     (1) 

required  of  the  working  force  P  for  steady  motion.  But,  con- 
sidering the  whole  collection  of  moving  "rigid"  bodies,  in- 
cluding each  particle  of  water,  but  ignoring  the  power  spent 
on  axle  friction,  and  considering  all  fluid  friction  to  be  en- 

rw  2 
tirely  represented  by  Q^xloss  of  head  -r^-  (see  §  107),  we  also> 

~~J 
have,  from  eq.  (15),  §  42a, 


(2) 


also,  wi  being  radial,              w\  =  V\  .........  (3) 

From  trigonometry, 

wJ-uJ+VJi   ........  (4) 

Vn  =  un  tan  /i,    ........  (5) 

V.n=(vn-Un)  tan  5,    ......  (6) 

and                                 cncosd  =  vn  —  un  ......     .  (7) 

The  equation  of  continuity  of  flow  is 

Q  =  2KrnenVn  =  27meiV1;    .....  (8) 

that  is,                                rnenVn  =  rieiVi  .......  (9) 

Also,                             vi  tan/?  =  wi,     .......  (10) 

V1  =  WTi,     .......  (11) 

and                                               vn=urn.    ../.....  (12) 

Combining  (1),  (2),  and  (4),  we  eliminate  P,  vf  Q,  7-,  and 
wn,  obtaining 

2gh  =  2unvn-(;(Vn2+ur?),   .....  (13) 

in  which,  if  for  Vn  we  substitute  its  value  (vn  —  un)  tan  d,  from 
(6),  there  results 


12 


178  HYDRAULIC    MOTORS.  §   109. 

Now  the  efficiency  rt  is  equal  to  the  ratio  of  the  portion  of 
power  applied  to  the  useful  purpose  of  raising  Qj-  Ibs.  of  water 
through  an  elevation  of  h  ft.  each  second,  to  the  whole  power, 
Pv,  exerted  by  the  working  force  per  second;  i.e., 


or  [see  eq.  (1)] 


The  value  of  h  from  (14)  may  now  be  substituted  in  (15), 
yielding 

-2+..    .    .    (16) 


Evidently  (16)  gives  the  efficiency  as  a  function  of  the  ratio 
+  Vn',    and  if  that 
(16)  may  be  written 


"Un  +  Vn',    and  if  that  ratio  be  denoted  by  x,  that  is,  if  x  =  —, 


>  ....    (17) 


which  is  a  function  of  but  one  variable,  x. 
By  differentiation, 


;     .    .    .    (18) 

the  placing  of  which  equal  to  zero  gives   the  special  value  of  jt 
(call  it  x')  which  makes  y  a  maximum,  viz., 

x'=         ===,     or    z'  =  sin£;       .     .     .     (19) 


i.e.,  for  a  maximum  efficiency  we  must  make 

un  =  vnsmd,    .......     (20) 

and  this  relation  substituted  in  eq.  (14)  gives,  after  considerable 
reduction,  the  value  of  vn  for  best  effect,  viz., 


v    — 


§  110.  CENTRIFUGAL   PUMPS.  179 

while   the  corresponding  maximum   efficiency  itself   becomes 
[see  eq.  (17)1 

t-i-ZZta+I (22) 

As  to  the  influence  of  the  choice  of  the  exit  vane-angle  d 
upon  this  expression  for  the  maximum  efficiency,  we  note 
that  the  latter  is  the  largest  possible,  viz.,  1.00,  when  £  =  0°. 
This  supposition,  however,  would  imply  a  zero  discharge, 
which  is  inadmissible.  It  would  also  make  the  corresponding 
vnf  =  infinity.  But  it  is  evident  that  d  should  be  taken  as 
small  as  practicable,  say  from  15°  to  30°.  If  d  were  as  great 
as  90°  (radial  tips)  and  the  friction  coefficient  £  as  large  as 
1.00  (which  would  doubtless  be  justified  if  the  casing  surround- 
ing the  pump  did  not  provide  a  gradually  enlarging  passage- 
way, with  guides,  for  the  water  leaving  the  pump),  we  should 
find  from  eq.  (22)  that  T/  is  only  about  0.50.  The  correspond- 
ing value  for  vn'  from  eq.  (21)  proves  to  be  vn'**\/2gh.  In 
fact,  Prof.  Zeuner  states,  in  his  book  on  "Theorie  der  Tur- 
binen,"  that  the  peripheral  speed  of  most  centrifugal  pumps 
in  regular  service  should  not  greatly  exceed  this  value,  v'2gh. 

That  a  greater  efficiency  is  obtained  from  impeller-blades 
curving  backward  as  in  Fig.  86,  as  against  that  obtained  when 
straight  radial  blades  are  used  (0  =  90°),  was  conclusively 
proved  by  actual  test  in  1851  by  Mr.  Appold,  who  introduced 
the  curved  blade. 

no.  Numerical  Example.  Centrifugal  Pump.  —  A  cen- 
trifugal pump  having  external  guides  ("  diffusion-guides ") 
providing  for  a  gradual  change  of  absolute  velocity  for  the 
water  as  it  leaves  the  impeller- blades  (and  hence  now  called  a 
"turbine  pump")  is  to  be  designed  for  a  head  of  h  =  36  ft.,  is 
to  pump  Q  =  50  cub.  ft.  per  sec.  in  steady  operation,  and  is  to 
work  at  an  angular  speed  of  300  revs,  per  min.  The  angle  d 
is  to  be  taken  =  30°  and  n  as  =}rn.  The  suction-  and  delivery- 
pipes  being  short  and  wide,  no  loss  of  head  will  be  considered 
as  occurring  in  them. 

Solution  (the  ft.-lb.-sec.  system  of  units  being  used). — Taking 


180  HYDRAULIC    MOTORS. 

a  value  of  0.5  for  £  from  the  favorable  conditions  provided 
by  the  guides  at  exit,  we  find  the  best  velocity  for  the  im- 
peller-tips to  be,  from  eq.  (21), 


I    32.2x36(1+0.5) 
^0.5  [1  +0.5  (1  -0.50)1"* 


•N0.5  [1+0.5  (1-0.50)]- 

To  find  rn  that  the  angular  speed  may  be  300  revs,  per  min.y 
we  write 

(300\ 
—  1=52.8;    whence    rn=1.68ft.; 

and  hence  n,  ='Jrn,  =0.84  ft. 

As  for  the  distance  between  crown-discs  (or  sides  of  the 
chamber,  if  there  are  no  crowns),  we  have,  from  eq.  (2)y 
un  =  Vn  sin  d;  that  is,  un  =  52.8X0.5  =  26.4  ft.  per  sec.;  and 
hence,  from  eq.  (6),  for  "  velocity  of  flow"  at  N, 

Vn-  (vn'-un)  tan  d=  (52.8-26.4)0.577, 
=  15.2  ft.  per  sec.;  and  hence,  finally,  from  eq.  (8), 

Q 


or  (say)  0.33  ft.  to  allow  for  the  thickness  of  the  (six  or  eight) 
impeller-blades;  i.e.,  ew  =  4  inches. 

In  order  to  secure  a  moderate  absolute  velocity  of  flow, 
Vi,  at  the  entrance  of  the  impeller  channels,  e\  may  be  as- 
sumed equal  to  3en,  i.e.  ,  =  1.00  ft.;  hence,  from  eq.  (9),  we  have 
the  "velocity  of  flow"  at  entrance,  Fi  =  fFn=10.1  ft.  per 
sec.,  which  also  =Wi,  since  the  latter  is  supposed  radial.  The 
necessary  value  for  the  vane-angle  at  entrance,  i.e.,  ft  follows;, 

viz.,  tan/?  =  —  =  —  *  r-r=  0.485;  or  8  must  be   taken 

Vi        Vi         %Vn        26.4 

as  (say)  26°.  A  smooth  curve  AN  (see  Fig.  86),  having  the 
proper  values  for  /?  and  d  at  its  extremities,  and  convex  on 
its  advancing  side  (as  in  that  figure),  will  serve  as  the  form  of 
the  (thin)  impeller-blade. 

As  to  the  efficiency  and  necessary  power  to  operate  the 
pump,  we  have  from  eq.  (22),  with  £  =  0.50  and  angle  £  =  30°, 


§   111.  CENTRIFUGAL    PUMPS.  181 

0.50 


which  may  be  called  the  "  hydraulic  efficiency/'  since  it  leaves 
out  of  account  the  power  spent  on  axle  friction  of  the  pump. 
Deducting  0.05  for  this  cause  we  obtain  0.78  as  the  value  of 
the  efficiency  from  which  the  necessary  power  is  to  be  com- 

puted.    Therefore,  placing  y'  =  ~p-;,  we  have  Pv=Qrh+y';  i.e., 

Pv=  (50X62.5X36)^0.78,  -144,200  ft.-lbs.  per  sec.;  or  262 
H.P.;  since  144,200-550  =  262. 

From  the  acknowledged  imperfection  of  the  theory,  these 
results  must  be  looked  upon  as  only  roughly  approximate. 
Much  experimentation  is  still  needed  to  supplement  the  de- 
ductions of  theory. 

in.  Practical  Points.  —  When  the  pump  is  situated  above 
the  source  of  supply,  T,  and  a  suction-pipe  is  therefore  neces- 
sary, its  elevation  above  T  is  of  course  restricted  (as  in  the 
case  of  the  draft-  tube  of  a  turbine)  to  a  value  considerably 
less  than  that  of  the  water-barometer  height.  In  such  a  case, 
when  the  pump  is  to  be  started,  it  is  found  impossible  by  the 
rotation  of  the  pump  itself  to  exhaust  the  air  from  the  suction- 
pipe.  This  must  first  be  done  by  closing  the  foot  -valve  at  the 
base  of  that  pipe  and  filling  up  with  water;  or,  after  closing 
a  valve  in  the  delivery-pipe,  to  exhaust  the  air  by  the  use 
of  a  steam-ejector,  as  is  frequently  done  when  a  steam-engine 
is  the  source  of  power,  the  water  being  thus  caused  to  rise  in 
the  suction-pipe  by  the  pressure  of  the  atmosphere  on  the 
lower  reservoir. 

If  the  suction-  and  delivery-pipes  have  considerable  length 
and  the  respective  losses  of  head  thus  occasioned,  when  the 
flow  Q  is  passing  through  them,  are  hf  and  h'"  respectively, 
and  if  the  water  is  delivered  in  a  free  jet,  of  w'  ft.  per  sec.  velocity, 
at  the  point  of  delivery,  then  the  h  of  the  preceding  theory 
wil  be  replaced  by 

w'2 
h+h'+hf"+—  .......     (23) 


182  HYDRAULIC    MOTORS.  §  112. 

It  amounts  to  the  same  thing  to  say  that  if  piezometer 
tubes  are  arranged  for  the  two  pipes,  at  points  near  the  pump 
(see  now  Fig.  5,  in  which  the  flow  of  water  must  be  conceived 
to  be  from  K  toward  A]  through  the  casing  M,  which  is  now 
supposed  to  contain  the  pump,  in  steady  operation),  then  the 
h  of  the  preceding  theory  is  to  be  replaced  by  the  h  of  Fig.  5, 

augmented  by  the  term  I  I ;    where  w%  is  the  (mean) 

L*9      *9  J 

velocity  of  the  water  passing  at  A,  and  w±  its  velocity  as  it 
passes  section  KH;  see  example  in  §  13. 

The  surfaces  'of  the  impeller-blades  should  be  as  smooth 
as  possible,  this  being  conducive  to  higher  efficiency.  Ex- 
periment has  shown  this  (see  Barr's  Pumping  Machinery, 
p.  343). 

112.  Centrifugal  Pumps  without  Gradual  Enlargement  Beyond 
Exit. — This  older  style  of  pump  has  been  found  to  give  fairly 
good  results  only  with  low  heads  (say  below  30  ft.),  the  high 
velocity  of  impeller  and  water  necessary  at  high  heads  causing 
a  large  amount  of  fluid  friction,  eddying,  etc.,  giving  rise  to 
large  losses  of  head.  A  good  example  of  the  ordinary  cen- 
trifugal pump  with  volute,  etc.,  but  without  external  guides, 
is  shown  in  Fig.  88  (the  Van  Wie  pump,  made  at  Syracuse, 
N.  Y.).  The  suction-pipe  is  attached  at  S  and  the  delivery- 
pipe  at  R.  E  is  a  steam-engine  furnishing  the  power  to  operate 
the  pump;  while  F  is  a  fly-wheel.  Pumps  of  this  type  have 
given  efficiencies,  under  low  heads,  as  high  as  65  per  cent., 
or  over. 

In  the  new  water-supply  system  of  Rockford,  111.,  designed 
and  carried  out  by  Prof.  D.  W.  Mead  in  1897,  three  cen- 
trifugal pumps  are  used,  constructed  by  the  Byron  Jackson 
Machine  Co.  of  San  Francisco,  which  gave  on  test  efficiencies 
of  from  70  to  75  per  cent.  Each  pump  worked  against  the 
same  head,  100  ft,,  of  which  26  ft.  was  "  suction-head/'  The 
impellers,  3.5  ft.  in  diam.,  are  of  bronze  and  have  carefully 
smoothed  interior  walls.  They  are  of  the  enclosed  type  (see 
§  114),  with  blades  curving  backward  (d  =  about  30°),  and 
have  "  dead  -spaces"  toward  the  outer  rim  between  water- 


FIG.  88. 


183. 


184  HYDRAULIC    MOTORS.  §   113. 

channels  (see  pp.  302  and  607  of  Turneaure  and  Russell's 
Public  Water-supplies;  also  p.  18  of  Engineering  News  for 
July  13,  1899).  A  valuable  article  by  Mr.  Richards  may  be 
found  in  vol.  xxxviii  of  the  Engineering  News,  pp.  75  and  91. 

113.  Turbine  Pumps.  Multi-stage  Pumps. — Within  a  few 
years*  centrifugal  pumps  have  been  constructed  in  f]urope, 
and  more  recently  in  America,  attaining  a  high  efficiency  under 
high  heads  by  the  use  of  gradually  enlarging  guide-passages 
receiving  the  water  immediately  on  exit  from  the  impeller 
channels,  thus  enabling  its  velocity  to  be  gradually  reduced 
from  the  value,  wn,  at  the  exit-point  of  impeller  to  the  slower 
velocity  of  the  delivery-pipe  or  other  passage  provided.  These 
.are  called  "turbine  pumps"  A  "multi-stage"  pump  consists 
of  a  series  of  two  or  more  impellers  on  the  same  shaft,  each 
pumping  water  into  the  central  space  of  the  next  adjoining 
(except  that  the  last  one  pumps  into  final  delivery-pipe),  the 
peripheral  pressure  of  one  being  therefore  nearly  equal  to  the 
•central,  or  receiving,  pressure  of  the  next.  The  intervening 
stationary  guide-passages  are  so  designed  as  to  produce  only 
.gradual  changes  in  the  absolute  velocity  of  the  water,  and  com- 
paratively high  efficiencies  are  thus  attained.  By  this  device  a 
high  head  (say  1000  ft.)  can  be  broken  up  into  steps,  as  it 
were,  each  impeller  having  to  deal  with  a  difference  of  pressure 
corresponding  to  the  fraction  of  the  whole  head  which  corre- 
sponds to  the  number  of  impellers. 

A  good  example  of  a  multi-stage  turbine  pump  is  shown  in 
Fig.  90  (which  gives  a  section  through  the  axis  of  rotation) 
and  Fig.  8')  (showing  a  section,  at  right  angles  to  the  shaft, 
through  one  of  the  four  impellers).  In  the  latter  figure  the 
walls  of  the  external  flaring  guide-passages  are  shown  in  solid 
black  shading.  The  impeller  is  seen  to  have  six  long  blades 
and  six  intervening  short  ones,  all  curving  backward  with 
respect  to  the  motion  of  rotation  (with  /?  and  d  each  =  about 
45°).  By  proper  passageways  the  water  is  conducted  from 
the  space  outside  of  an  impeller  to  the  central  space  of  the  next 

*See  Mr.  Webber's  article  in  Cassier's  Magazine  for  June  1905,  p.  154. 


Transverse  Section 
FIG.  89. 


Longitudinal     Section. 
FIG.  90. 


185 


186  HYDRAULIC    MOTORS.  §  113- 

one  of  the  series  and  finally  into  the  delivery-pipe  (for  detailed 
explanation,  see  Engineering  News,  Jan.  1902,  p.  66).  The 
diameter  of  each  impeller  is  20  in.  and  (on  test)  water  was  pumped 
at  the  rate  of  Q  =  2A7  cub.  ft.  per  second  against  a  head  of 
425  ft.,  the  pump  rotating  at  890  revs,  per  min.  Each  im- 
peller therefore  had  to  pump  against  a  difference  of  pressure 
corresponding  to  a  head  of  106  ft.  In  the  same  test  the  effi- 
ciency was  found  to  be  76  per  cent.  This  pump  was  designed 
and  constructed  by  the  firm  of  Sulzer  Bros,  of  Winterthur, 
Switzerland. 

On  p.  324  of  Engineering  News  for  April  7,  1904,  may  be 
found  an  illustrated  article  describing  several  "  High-pressure 
Multi-stage  Turbine  Pumps"  built  by  the  Byron  Jackson 
Machine  Works  of  San  Francisco,  California.  (Quoting  from 
this  article :)  "  Pumps  of  this  design  are  built  for  heads  of  from 
100  to  2000  ft.,  the  number  of  separate  impellers  or  l  stages ' 
being  properly  proportioned  to  the  head.  About  100  to  250  ft. 
head  per  stage  appears  to  be  allowed."  A  two-stage  pump 
built  for  the  water-works  of  the  city  of  Stockton,  Cal.,  delivers 
1500  gallons  per  minute  against  a  head  of  140  ft.  at  690  revs, 
per  min.  The  pump  was  guaranteed  to  have  an  efficiency 
of  at  least  75  per  cent.,  but  developed  82  per  cent,  at  the  official 
test 

Since  the  water  is  usually  admitted  to  the  central  impeller 
space  from  one  side  only,  an  end  thrust  of  the  shaft 
against  its  bearings  is  thereby  created  unless  prevented 
by  special  device  .  In  Fig.  91  is  shown  a  section  (through 
axis  of  shaft)  of  a  6-in.,  six-stage,  " spherical,"  "compound 
pump"  (i.e.,  multi-stage  pump)  built  by  the  Lawrence  Machine 
Co.  of  Lawrence,  Mass.,  and  so  constructed,  by  the  arrange- 
ment of  the  impellers  in  pairs  and  by  the  position  of  the  inter- 
vening guide-passages,  that  the  resultant  end  thrust  is  zero. 
To  quote  from  the  printed  circular:  "These  pumps,  like  all 
of  this  type,  are  provided  with  diffusion- vanes  directly  at  t he- 
periphery  of  the  impellers,  and,  unlike  others  of  their  type, 
the  liquid  is  not  forced  through  short  tortuous  passages  imme- 
diately after  passing  through  the  diffusion- vanes,  before  enter- 


FIG,  91.     Six  Stage  "Spherical"  Compound  Pump. 
Made  by  the  Lawrence  Machine  Co. 


§  114.  CENTRIFUGAL   PUMPS.  187 

ing  the  next  successive  impellers,  but  instead  through  long  easy 
passages  of  uniform  cross-section  and  easy  curves. "  The  term 
" spherical"  is  due  to  the  outside  appearance  of  the  pump-case. 

114.  Practical  Notes. — Since  there  are  no  valves  or  other 
moving  parts  in  a  centrifugal  pump,  except  the  impeller  itself, 
this  type  of  pump  is  admirably  adapted  for  the  pumping  of 
viscous  and  stringy  liquids,  or  liquids  containing  sand  or  silt 
in  suspension,  or  even  carrying  chips,  bark,  and  gravel. 

The  large  hydraulic  dredges  used  by  the  Mississippi  River 
Commission  pump  the  river- water,  charged  with  silt  or  sand 
by  previous  stirring  of  the  botfom,  through  long  pipes  to  a 
" spoil-bank"  at  some  distance,  thereby  deepening  the  channel 
for  purposes  of  navigation.  (See  reference  in  §  13;  also  En- 
gineering News,  Oct.  1898,  p.  236.) 

Another  advantage  is  that,  since  the  centrifugal  pump  is  a 
body  rotating  continuously  in  one  direction,  the  shaft  may  be 
coupled  directly  to  that  of  an  electric  motor.  A  pump  having 
crowns  or  discs  forming  part  of  the  impeller  is  said  to  be  of 
the  "  enclosed  "  type.  If  it  consists  merely  of  the  impeller-blades, 
fastened  to  and  projecting  from  a  spindle  or  shaft,  it  is  called 
"  unenclosed."  In  this 'latter  case  the  stationary  sides  of  the 
pump- case  serve  as  crowns,  the  edges  of  the  impeller- blades 
revolving  almost  in  contact  with  them. 


CHAPTER  VIII. 
PIPES,  WEIRS,  AND  OPEN  CHANNELS. 

(Note.  —  This  chapter  contains  matter  supplementary  to 
Chapters  VI  and  VII  of  the  writer's  Mechanics  of  Engineering. 
Bernoulli's  Theorem  for  steady  flow  of  water  in  (rigid,  station- 
ary) pipes  and  stream-  lines  is  already  proved  in  §§  492  and  512 
of  that  work.) 

115.  Friction-head  in  Long  Pipes.  —  Since  long  pipes  and 
penstocks*  are  frequently  used  to  convey  water  to  hydra  L  lie 
motors,  the  'loss  of  head  so  occasioned  is  an  important  con- 
sideration. For  a  steady  flow  of  water  in  a  stationary  rigid 
pipe  of  cylindrical  form  the  loss  of  head  due  to  fluid  friction 
(see  eq.  (4),  p.  700,  M.  of  E.)  is  conveniently  expressed  in  the 
form  4fl  0 

kF  =  ^'2g'       '.   ......     (1) 

where  I  is  the  length  and  d  the  internal  diameter  of  the  pipe,  v 
the  mean  velocity  (component  parallel  to  axis  of  pipe)  of  the 
particles  of  water  passing  through  any  given  cross-section  (gen- 
erally about  83  per  cent,  of  the  velocity  of  particles  near  the 
center  of  the  section)  and  /  a  "  coefficient  of  fluid  friction," 
or  abstract  number,  to  be  determined  by  experiment. 

The  volume  of  water  flowing  per  second  is,  of  course,  Q=Fv, 


For  new  and  clean  cast-iron  pipes,  and  for  such  small  sizes 
of  wrought-iron  pipe  as  involve  no  riveting,  Mr.  Fanning's 
tables  of  values  for  the  coefficient  give  fairly  trustworthy 
results;  but  much  time  may  be  saved  by  the  use  of  diagrams 

*  For  articles  on  penstocks  see  pt  549  of  the  Engineering  Recoid  of  Nov. 
14,  190S;  and  p.  172  oi'  the  Engineering  News  of  Feb.  10,  1910. 

188 


§  115.  PIPES    AND    PENSTOCKS.  1891 

which  enable  the  friction-head  itself  to  be  found  with  great 
directness.  Of  course  in  such  a  case  it  makes  no  difference 
whether  the  formula  upon  which  the  diagram  is  based  is  simple 
or  complicated.  The  diagrams  prepared  by  the  present  writer 
for  pipes  of  above  description,  founded  on  Mr.  Fanning's 
values  for  /,  have  been  placed  in  the  Appendix  of  this  work. 
Results  obtained  from  these  diagrams  will  be  found  to  differ 
but  slightly  from  those  based  on  Mr.  Metcalfe's  "Diagram  D," 
published  in  the  Engineering  Record  of  June  20,  1903.  This 
diagram  is  stated  by  Mr.  Metcalfe  to  be  "  for  general  use  with 
new  cast-iron  pipes"  and  is  based  on  the  Hazen- Williams  for- 
mula, 68 

(mean  velocity)  v=71.6d*™sr™;       ...     (2) 

in  which  s  denotes  the  ratio  -4-  and  the  foot  and  second  are 

to  be  used  as  units.  (For  an  account  of  the  Hazen-Williams 
hydraulic  slide-rule,  see  the  Engineering  Record  for  March  28  > 
1903.) 

With  increasing  age  of  service  cast-iron  pipes  are  liable  to 
become  corroded  and  tuberculated  (if  originally  tar-coated  this 
action  may  be  much  retarded) ,  which  diminishes  the  discharge 
under  the  same  head  (both  from  increased  roughness  and 
diminished  sectional  area). 

Mr.  E.  B.  Weston  recommends  that  for  pipes  of  cast-iroa 
the  friction-head  for  a  given  Q  be  taken  as  16  per  cent,  greater 
than  when  the  pipe  is  new  and  clean,  for  each  five  years  of  age. 
For  example,  for  an  age  of  15  years  take  as  the  friction-head 
for  a  given  flow  Q,  and  per  1000  ft.  of  length,  the  value  obtained 
by  multiplying  the  result  given  by  the  diagram  by  1.48.* 

According  to  the  recommendations  of  Mr.  Metcalfe  (see  above 
article),  we  may  find  the  friction-head  hp  for  old  and  tuber- 
culated pipe  for  a  given  mean  velocity  by  taking  -f$f  of  that 
given  by  the  diagram  for  clean  cast-iron  pipes  for  the  same 
velocity;  or,  to  put  it  another  way,  for  a  given  friction- head 
the  velocity  obtained  from  the  diagram  for  clean  cast  iron  pipe 
must  be  multiplied  by  |f  to  give  the  velocity  for  the  tuber- 

*  A  useful  book  in  this  connection  is  "  Hydraulic  Tables,"  by  Prof.  G.  S, 
Williams  and  Mr.  Allen  Hazen  (New  York:  John  Wiley  &  Sons,  1905). 


190  HYDRAULIC    MOTORS.  §   116. 

culated    pipe.     However,    considerations    of    friction-head    in 
old  and  tuberculated  pipes  involve  much  uncertainty. 

Similarly,,  according  to  Mr.  MetcahVs  article,  in  the  case 
of  riveted  iron  and  steel  pipe,  the  coefficient  /  is  so  increased 
(on  account  of  the  projecting  rivet-heads,  etc.)  that  a  value 
of  the  friction-head  taken  from  a  diagram  for  clean  cast-iron 
pipe  must  be  multiplied  by  -J-Q-J-  to  give  that  for  the  riveted  pipe 
for  the  same  diameter  and  velocity;  or,  conversely,  if  the 
value  of  the  mean  velocity  has  been  taken  from  the  diagram 
for  clean  cast-iron  pipe  for  a  given  diameter  and  friction-head, 
it  must  be  multiplied  by  -?  to  give  the  proper  velocity  for  the 
riveted  pipe.* 

116.  Conversion  Scales. — From  the  fact  that  great  nicety  is 
useless  in  computations  for  hydraulic  problems  involving 
friction-heads,  it  is  sufficiently  accurate  in  most  cases  to  use 
values  taken  from  diagrams;  to.  expedite  the  work  (and,  in- 
cidentally, to  avoid  gross  errors). 

In  the  Appendix  to  this  work  will  be  found  a  page  of  "con 
version  scales/'   by  the  use  of  which  the  velocity-head,  hv, 
corresponding  to  a  given  velocity,  v,  may  be  found,  and  vice 
versa;    the  hydrostatic  pressure,  p,  in  Ibs.  per  sq.  in.,  due  to 

79 

a  "pressure-head,"  or  static  head,  h  =  -,  of  water,  in  feet;  or 

P 
the  pressure-head,  — ,  in  feet,  corresponding  to  a  given  pressure, 

p,  in  Ibs.  per  sq.  in.,  etc.;  and  scales  for  converting  a  discharge, 
Q,  in  cub.  ft.  per  second,  into  gallons  per  minute;  etc.,  etc. 

The  quantities  involved  in  any  two  adjoining  scales  are 
directly  proportional  to  each  other  except  in  the  case  of  the 
velocity-scale,  where  the  velocity-head  hv  is  proportional  to  the 
square  of  the  velocity  v.  In  the  use  of  the  velocity-scale, 
therefore,  this  relation  must  be  borne  in  mind  in  dealing  with 
values  that  extend  beyond  the  limits  of  the  scales.  For  ex- 
ample, if  the  velocity-head  hv  for  a  velocity  of  i>=120  ft.  per 
sec.  is  desired,  find  the  hv  for  one  half  of  120  (i.e.,  for  60)  or 
56  ft.,  and  multiply  by  4,  which  gives  224  ft;  and,  again,  if 
we  wish  the  velocity  corresponding  to  an  hv=lSO  ft.,  we  first 

*  A  special  diagram  giving  friction-heads  for  riveted  steel  pipe  has  now 
(1911)  been  added  to  the  Appendix.     See   eference  on  diagram. 


§   118.  PIPES    AND   PENSTOCKS.  191 

find  the  v  (or  35.8)  for  20  ft.,  which  is  one-ninth  of  180,  and 
multiply  by  3,  obtaining  107.4  ft.  per  second  (or  find  v  (  =  53.8) 
for  one- quarter  of  hv  and  multiply  by  2) . 

117.  The     Hydraulic    Grade-line. — This    has    been    defined 
(see  p.  715,  M.  of  E.)  as  the  line  containing  the  summits  of  the 
stationary  water  columns  in  the  open  piezometers  that  may 
be  imagined  to  be  placed  at  various  points  along  a  pipe  in 
which   water  is  flowing  in  "steady  flow."     Along  a  straight 
pipe  of  uniform  diameter  this  line  is  straight  and  slopes  down- 
ward for  points  farther  and  farther  down-stream  (the  slope 
of  the  pipe  itself  is  immaterial).     The  reason  for  this  inclined 
position  is  the  friction-head  along  the  pipe;  if  this  were  zero, 
the  grade-line  would  be  horizontal.     But  if  a  portion  of  pipe 
has   a  decreasing  sectional  area  (going   down-stream)   (e.g.,  a 
conically  converging  pipe),  the  grade-line  drops  more  rapidly 
on  account  of  the  increase  in  velocity-head  in  successive  cross- 
sections;    and,  conversely,  along  a  portion  of  the  pipe  which 
is  conically  divergent  the  grade-line  rises  (unless  the  divergence 
is  so  slight  that  the  rise  due  to  decrease  in  velocity-head  is 
offset  by  the  drop  due  to  friction-head).     All  of  these  state- 
ments  are   easily   proved   by   the   application   of    Bernoulli's 
Theorem  to  the  two  extremities  of  any  given  portion  of  the 
pipe.     A  few  numerical  examples  will  now  be  worked  out  in 
illustration  of  the  conception  of  the  hydraulic  grade-line  and 
also  of  the  use  of  the  friction-head  diagrams  (Appendix). 

118.  Numerical  Problems.     (I)  Single  Pipe;   without  Nozzle. 
—Fig.  92..    A  steady  flow  of  water  is  taking  place  through 
the   horizontal   cylindrical   pipe   (clean   cast-iron   pipe),  whose 
length  is  80  ft.  and  diameter  4  in.,  from  the  large  reservoir  R. 
The  entrance  of  the  pipe  at  E  is  not  rounded.      The  head 
7i  =  9.3  ft.     There  is  no  nozzle  at  the  end  m  of  the  pipe,  so  that 
at  that  point  the  jet  entering  the  atmosphere  has  the  same 
sectional  area  as  the  pipe  and  a  mean  velocity  vm  equal  to 
that,  ?',  in  the  pipe.     At  any  point,  such  as  S,  (not  nearer  than 
12  inches  to  the  side  of  reservoir,)  if  the  length  ES  =  x,  we  find, 
by  applying  Bernoulli's  Theorem  between  the  point  S  of  flow 
.and  the  surface  of  (still    water  in  R,  that  the  height  of  the 


192  HYDRAULIC    MOTORS. 

open  piezometer  at  S  is 

PS,  -y,  =  h-~-     * 


118. 


(3) 


where  hx  is  the  loss  of  head  due  to  skin  friction  along  length 
ES  of  pipe,  and  equals  the  vertical  drop  from  B  to  P;    and 

v2 
£E--  is  loss  of  head  at  entrance  E  (see  pp.  706  and  711,  M. 

of  E.). 

Now  hx  is  proportional  to  x  and  becomes  =  hp  or  vertical 
drop  from  B  to  m  when.  x=  whole  length  I,  hF  being  the  friction- 


Rl 


p 

r  > 
ts 


•    FIG.  92. 

head  for  whole  length  of  pipe.     Here  (no  nozzle)  the  velocity 
v=vm,   and  is  to   be  determined.     Therefore  7/=zero  at   m; 

v2 


and 


(4) 


The  whole  h  is  seen  to  be  made  up  of  three  parts,  of  which,  in 
this  case  hF  (there  being  no  nozzle*)  will  probably  be  nearly 
equal  to  h  itself.  We  shall  now  solve  by  trial,  using  the  friction- 
head  diagrams  in  Appendix  for  clean  cast-iron  pipe. 

Assume  as  a  first  trial  value  that  hp  =  7  ft.  This  is  at  the 
rate  of  (7-*- 0.080  =  )  87.5  ft.  Motion-head  per  1000  ft.  of  pipe 
length.  Turning  now  to  the  diagram  for  the  smaller  pipes 
(J  to  8  in.  in  diameter),  we  find  the  vertical  line  corresponding 
to  87.5  among  the  figures  along  the  upper  edge  and  note  its 

*  And  the  length  of  pipe  being  large  compared  with  its  diameter;  240 
''diameters"  long. 


§  119.  HYDRAULIC    GRADE-LINE.  193 

intersection  with  the  oblique  line  marked  "4-in.  pipe."  Among 
the  other  oblique  lines  (velocity  lines)  this  intersection-point 
corresponds  to  a  velocity  of  9.1  ft.  per  sec.,  for  which  (see 
highest  scale  on  page  of  "  Con  version  Scales,"  Appendix)  the 
velocity-head,  or  v2  +  2g,  =1.3  ft.  Since  £#  =  0.50,  or  J,  the 
loss  of  head  at  E  is  i(1.3)  =0.65  ft.  Hence  the  sum 

0.65 +  7 +  1.3  =  8.95  ft. 

But  this  lacks  0.35  ft.  of  what  it  should  be,  viz.,  9.30  ft. 
For  the  next  trial  it  will  probably  occasion  no  great  error  if 
the  whole  of  this  0.35  be  added  to  the  original  7  ft.  That  is, 
assume  hF  =  7.35  ft.,  which  is  at  the  rate  of  (7.35^0.080  =  ) 
92  ft.  friction-head  per  1000  ft.  of  pipe.  The  diagram  now 
gives  v=9.4  ft.  per  sec.  in  a  4-in.  pipe,  and  the  velocity-head 
=  1.4  ft.  and  i  of  1.4  =  0.7  ft.  Adding,  we  have  0.7  +  7.35  + 1.4 
=  9.45  ft.,  which  is  so  near  to  the  required  9.3  ft.,  or  h,  that 
this  second  trial  may  be  considered  final.  The  corresponding 
discharge  is  Q  =  0.81  cub.  ft.  per  sec.  (found  by  following  a 
horizontal  line,  through  the  intersection  of  the  vertical  92  and 
the  4-in.  pipe  line,  to  the  scale  on  right-hand  edge  of  diagram). 

In  Fig.  92  the  vertical  distance  AB  is  the  sum  of  the  en- 
trance loss  of  head  and  the  velocity-head  v2  +  2g.  In  the 
contracted  vein  at  E  the  pressure-head  is  less  (velocity  being 
much  greater)  than  for  the  point  under  B,  which  is  about 
three  diameters  or  12  inches  from  the  side  of  reservoir.  Most 
of  the  entry  loss  of  head  occurs  between  the  neck  of  the  con- 
tracted vein  and  a  point  under  B,  but  it  is  less  than  the  differ- 
ence of  velocity-heads  at  that  point  and  B. 

Note. — If  the  jet  discharges  under  water,  the  results  are 
the  same  provided  the  surface  of  the  water  in  the  receiving- 
reservoir  is  9.3  ft.  below  that  in  the  supply-reservoir  R  (both 
reservoirs  large) . 

It  is  immaterial  whether  the  pipe  is  horizontal  or  not,  if 
Z  =  80  ft.  and  ft  =  9.3  ft. 

119.  Numerical  Problems.  (II)  Single  Pipe;  with  a  Nozzle. 
(Fig.  93.) — Clean  cast-iron  pipe  of  6  in.  diameter,  1600  ft. 
long;  with  a  gradually  tapering  nozzle  (or  "  play-pipe,"  for  a 


194  HYDRAULIC    MOTORS.  §  119. 

fire-stream).  At  the  tip  of  the  nozzle  the  water  forms  (in  the 
atmosphere)  a  jet  with  parallel  filaments  (no  contraction)  and 
a  diameter  dm  =  2  inches.  The  head  h  is  90  ft.  and  the  loss 
of  head  in  the  nozzle  may  be  taken  as  0.05  (or  1/20)  of  vm2  +  2g, 
where  vm  is  the  velocity  of  the  jet;  *  while  the  entry  loss  of  head 
at  E  (corners  not  rounded)  is  ^  of  v2  -r-  2g,  v  being  the  velocity 
of  the  water  in  the  6-in.  pipe.  From  the  equation  of  continuity, 
the  pipe  running  full,  and  the  flow  having  become  steady, 
we  have  vm  =  9v.  It  is  required  to  find  the  two  velocities 
v  and  vm  and  the  discharge  Q;  use  being  made  of  the  friction- 
head  diagrams  (Appendix). 

Solution.  —  Bernoulli's  Theorem  applied  between  reservoir 
surface  A  (R  is  a  large  reservoir,  so  that  velocity  at  A  is  taken 
as  zero),  note  being  made  that  the  water-barometer  height,  6, 
cancels  out  (occurring  in  the  expression  for  pressure  head  both 
at  A  and  at  m),  gives 

1  v2  1    vm2    Vm2  , 


hF  denoting  the  loss  of  head  in  the  6-in.  pipe. 

We  here  note  that  the  whole  head  h  is  made  up  of  four 
items,  viz.,  three  losses  of  head  and  the  velocity-head  in  the 
free  jet  (the  student  will  note  the  corresponding  vertical  heights 
in  Fig.  93).  In  this  case  hp  is  not  necessarily  a  large  portion 
of  hj  since  there  must  be  considerable  pressure-head  (  =  DE'  +b) 
at  E',  the  base  of  the  nozzle,  to  account  for  the  great  change 
of  velocity  between  E'  and  m.  We  now  solve,  by  the  use  of 
the  proper  friction-head  diagram  (containing  the  6-in.  size  of 
pipe)  by  successive  assumptions  for  the  smaller  velocity,  v. 

First  assume  v  =  5  ft.  per  sec.,  for  which  from  the  diagram 
(for  6-in.  pipe)  we  find  the  friction-head  would  be  at  the  rate 
of  18  ft.  per  1000  ft.  of  length,  and  hence  hF  would  be  £££§-  of 
18  ?  =28.8  ft.  From  velocity-head  scale  (above),  since  v=-5 
and  rm  =  9X5=45,  we  obtain  v2/2g  =  QAQ  ft.  and  vm2/2g= 
31.4  ft.  Hence  the  two  small  losses  of  head  would  be  one  half 

*  See  foot-note  on  p.  70G,  M.  of.  E. 


119. 


NUMERICAL   PROBLEMS.      PIFE3. 


195 


Of  o.4,  =0.2;  and  1/20  of  31.4  =  1.57  ft.     The  sum  of  these  is 
0.2  +  28.8  +  1.57  +  31.4,  =61.97  ft.  (but  it  should  be  90  ft.). 

Second  Trial. — Take  v  =  6  ft.  per  sec.  vm  would  be  54  ft. 
per  sec.,  and  the  two  velocity-heads  would  be  0.56  and  45  ft.; 
hence  the  two  small  losses  of  head  are  0.28  and  ^  of  45, 
=  2.25  ft.  Now  6  ft.  per  sec.  in  a  6-in.  pipe  implies  a  friction- 
head  at  rate  of  25  ft.  per  1000  ft.  of  length  and  hence  hp  would 
of  25,  =40  ft.  Forming  the  sum,  we  have  0.28+40 


+  2.25+45,    =87.53   ft.     the   difference   between   which   and 


l  =  16OO    FT. 


FIG.  93. 


90  ft.  is  so  small  that  a  value  of  6.1  ft.  per  sec.  may  be  con- 
sidered as  a  final  solution  for  v;  from  which  follow  the  values 
55  ft.  per  sec.  for  vm  and  (see  diagram)  1.2  cub.  ft.  per  sec. 
for  the  discharge,  Q;  Ans. 

With  increasing  age  the  discharge  and  (v)  would  of  course 
gradually  diminish  unless  the  pipe  were  kept  clean.  If  the 
entrance  E  were  rounded,  a  slight  increase  of  Q  would  result. 

These  values  of  the  jet  velocity  vm  and  discharge  Q  are  the 
same  as  if  the  nozzle  or  play-pipe  issued  from  the  vertical  side 
of  a  large  tank  containing  water  the  height  of  whose  upper 

V2 

surface  above  the  point  E'  is  DEr  +^-;    (proved  by  applying 

Bernoulli's  Theorem  to  the  base  of  nozzle  as  up-stream  position 
and  m  as  down-stream  position).  In  the  present  case  v2  +  2g 
is  small,  only  0.60  ft.  The  height  DE'  of  piezometer  at  E'  is 


Q/x 


FIG.  94. 


*  30,000' 

Q    =    33  CUB.  FT.  PER  SEC. 

Q'=     13  CUB.  FT,  PER  SEC. 

Q"=     20  CUB.  FT.  PER  SEC.  DlAMETERS=? 


95- 


196 


§  120.  NUMERICAL    PROBLEMS.      PIPES.  197 

easily  found  to  be  48.7  ft.;   and  the  pressure  at  base  of  nozzle 
is  therefore  21  Ibs.  per  sq.  in.  (see  Conversion  Scales,  Appendix) . 

120.  Variation  in  Last  Problem.     (Fig.  93.) — Instead  of  the 
diameter  of  the  pipe  being  given,  let  us  inquire  what  should  be 
its  value  in  order  that  80  ft.  of  the  total  90  ft.  head  may  be 
available  to  produce  the  jet  velocity  vm;    that  is,  that  only 
10  ft.  of  the  90  ft.  may  be  lost  in  friction-head  and  the  two 
entrance  losses  of  head;   the  remainder,  80  ft.,  being  =  vm2/2g. 
In  this  case  vm  itself  would  be  71.8  ft.  per  sec. 

In  the  nozzle  the  loss  of  head  would  be  1/20  of  80  ft.;  i.e., 
4  ft.;  while  that  at  E  may  be  neglected.  This  leaves  10  —  4, 
=  6  ft.,  for  hF,  which  is  at  the  rate  of  (6/1.6=)  3.75  ft.  per  1000 
ft.  of  length.  Now  a  jet  of  71.8  ft.  per  sec.  velocity  and  of  2  in. 
diameter  is  discharging  Q=1.56  cub.  ft.  per  sec.  [obtained  by 
multiplying  71.8  by  the  area  (sq.  ft.)  of  a  2-in.  circle;  or,  more 
simply,  by  the  friction-head  diagram  (one  quarter  of  71.8  is 
.18  (say) ,  which  is  within  the  limits  of  diagram  and  for  a  2-in. 
area  gives  Q  =  0.39,  which  multiplied  by  4=  1.56)]. 

With  the  3.75  and  Q=1.56,  we  find  from  diagram  that  a 
diameter  of  9.9  inches  (say  10  in.)  must  be  given  to  the  pipe 
in  Fig.  93.  Ans. 

This  change  of  design  calls  for  a  greater  consumption  of 
water  (1.56  instead  of  1.20  cub.  ft.  per  sec.),  but  the  "kinetic 
power"  of  the  "free  jet"  at  m  (that  is,  the  kinetic  energy  of 

s~\  f\ 

the  mass  flowing  per  sec.  in  jet),  viz., ~~,  will   be   more 

9     A 

than  doubled.     It  will  be  7800  ft.-lbs.  per  sec.  instead  of  3513; 
i.e.,  14.2  H.P.,  instead  of  6.4. 

As  another  variation  (for  the  student  to  work  out) :  Given 
Q,  h,  d,  and  Z,  determine  necessary  values  for  vm  and  dm  to 
realize  this  discharge.  Also  find  the  H.P.  of  jet  and  the  power 
to  be  expected  from  a  Pelton  wheel  of  80  per  cent,  efficiency. 

121.  Main  Pipe  and  Two  Branches.  (Fig.  94.) — A  steady  flow 
of  water  is  to  take  place  from  reservoir  R  to  two  lower  reservoirs, 
Rf  and  R",  through  a  main  pipe  EP  and  two  branch  pipes, 
JA7'  and  JN",  each  of  which  discharges  under  water  (at  N' 
and  N"  respectively) .     No  nozzles  are  provided,  so  that  the  ve- 


198  HYDRAULIC   MOTORS.  §  121. 

locity  of  each  submerged  jet  is  equal  to  that  in  the  branch  pipe 
itself,  and  the  hydraulic  grade-line  for  each  branch  is  a  straight 
line  from  the  junction  /  to  points  U  and  L"  in  the  receiving- 
reservoirs  vertically  over  the  discharging  ends  of  the  pipes. 
The  flow  having  adjusted  itself  to  a  "  steady"  condition,  the 
flow  in  EP  of  Q  cub.  ft.  per  sec.  will  be  equal  to  the  sum  of 
those,  Q'  and  Q",  in  the  two  branches.  If  a  piezometer  were 
inserted  just  above  the  junction,  J,  the  summit  of  the  station- 
ary water  column  therein  would  be  at  some  point  C  in  the 
tube,  and  the  straight  line  BC  is  the  hydraulic  grade-line  for 
EP.  Similarly  D'L'  would  be  the  (straight),  hydraulic  grade- 
line  for  pipe  JR1  ',  Df  being  vertically  over  a  point  in  the 
pipe  where  the  loss  of  head  due  to  skin  friction  proper  begins; 
there  being  a  local  loss  (like  that  for  an  elbow)  at  the  junc- 
tion, and  also  a  change  of  velocity,  for  those  stream-lines 
which  enter  this  branch.  A  corresponding  statement  may 
be  made  for  the  other  branch. 

Let  v,  vf,  and  v"  be  the  velocities  of  steady  flow  in  the  three 
pipes,  respectively;  and  their  lengths  and  diameters,  and  the 
elevations  of  reservoirs,  be  as  indicated  in  Fig.  94.  The  friction- 
head  hp  for  pipe  EP  is  the  vertical  projection  of  its  hydraulic 
grade-line.  Similarly  hp'  nad  hF"  are  the  friction-heads  of 
the  branch  pipes.  As  to  the  other  vertical  "  drops"  between 
A  and  Lf,  and  A  and  L"  ,  we  have  (from  Bernoulli's  Theorem) 

v2     v2  v'2 


and 


v'2  v"2 

in  which  £'y   and  C"~7T~  are  losses  of  head  due  to  change  of 

section  (if  abrupt)  or  elbow  resistance. 

Now  in  most  cases  in  practice  the  velocities  in  the  pipes  of 
a  system  are  rarely  over  10  ft.  per  second,  and  the  pipes  are 
very  long  (as  in  next  paragraph)  ;  so  that  in  treating  a  problem 
like  the  present  (one  where  the  Q's  are  required  if  the  diameters 


§  122.  NUMERICAL   PROBLEMS.      PIPES.  199 

are  given,  or  vice  versa) ,  it  is  sufficiently  accurate  to  neglect  the 
small  "  drops^  AB,  CD',  and  CD"  in  the  hydraulic  grade-lines 
and  consider  that  the  whole  drop  from  surface  of  water  in  R  to 
that  in  Rf  is  equal  to  the  sum  of  the  two  friction-heads  hp  and. 
hp';  and  similarly  that  the  drop  from  R  to  R"  =  hp  +  hp"» 
(However,  this  would  not  be  justified  if  there  were  nozzles  at 
N'  and  N";  see  Fig.  93.) 

Problems  of  this  kind  are  best  solved  by  trial,  use  being 
made  of  friction-head  diagrams.  Other  modes  of  solution  are 
very  tedious  and  intricate. 

122.  Numerical  Problems.  (Ill)  Main  Pipe  and  Two 
Branches. — For  the  system  of  pipes  in  Fig.  95  (same  as  in  Fig. 
94,  but  with  numerical  data) ,  such  diameters  are  to  be  determined 
for  the  three  pipes  respectively  that  the  discharge  shall  be 
Q=33  cub.  ft.  per  sec.  through  the  main  pipe,  of  which  (Q'  =  ) 
13  is  to  pass  to  reservoir  R'  and  (Q"  =  )  20  to  R".  Elevations 
and  lengths  are  as  printed  in  Fig.  95.  (Clean  cast-iron  pipes.) 

We  are  at  liberty  to  assume  one  of  the  diameters;  or  the 
friction-head,  hF,  of  the  main  pipe;  say  the  latter.  Take  hp=4Q 
ft.  A  steady  flow  is  to  take  place  in  pipe  EJ  of  33  cub.  ft.  per 
sec.  and  the  friction-head  is  to  be  at  rate  of  (40-^30  =  )  1.33  ft. 
per  1000  ft.  of  length.  In  the  diagram  of  friction-heads  for 
large  pipe  (see  Appendix)  we  note  that  the  vertical  line  for  1.33 
(interpolating)  intersects  the  horiz.  line  for  Q=33  in  a  point 
corresponding  to  a  diameter  of  38  in.  (among  the  lines  sloping 
up  to  the  right) ,  while  among  the  other  inclined  lines  (sloping 
down  to  the  right)  we  find  that  with  this  discharge  the  velocity 
of  the  water  in  this  38-in.  pipe  would  be  4.1  ft.  per  sec.  (which 
is  not  extreme).  Deducting  the  assumed  hp  (40  ft.)  from 
the  altitude  60  ft.,  we  find  the  corresponding  value  of  hp'  to 
be  20  ft.;  i.e.,  at  the  rate  of  (20 -s- 10  =  )  2  ft.  per  1000  ft.  length 
of  pipe.  From  same  diagram  we  note  that  the  intersection  of 
the  vertical  2  with  the  horizontal  for  Q=  13  is  a  point  calling 
for  a  25-in.  pipe;  in  which  with  this  value  of  Q  (13)  the  velocity 
of  the  water  would  be  3.8  ft.  per  sec.  (a  permissible  value). 

Similarly,  deducting  the  hp  (40  ft.)  from  the  85  ft.  altitude 
we  obtain  for  the  hp"  of  the  other  branch  pipe  45  ft. ;  which  is 


200  HYDRAULIC   MOTORS.  §  123. 

at  the  rate  of  (45*  15  =  )  3  ft.  friction-head  per  1000  ft.  of 
length;  for  which,  with  Q=20,  the  diagram  gives  a  diameter 
of  27.5  in.  for  pipe  JN"  with  a  velocity =  5  ft.  per  sec. 

If  hp  had  been  assumed  somewhat  >  than  40  ft.,  a  smaller 
diameter  would  have  resulted  for  the  main  pipe,  EJ,  with  a 
higher  velocity  in  it  than  before;  but  larger  diameters  and 
smaller  velocities  in  the  two  branch  pipes.  Results  should  be 
sought  involving  the  least  cost,  with  sufficient  velocities  (above 
2  ft.  per  sec.)  to  prevent  the  deposit  of  silt. 

123.  Variation    from   Foregoing  Problem.  —  In    the    above 
example  the  diameters  were  the  quantities  sought;   but  if  the 
diameters  were  given  and  the  rates  of  flow  that  would  occur  in 
the  respective   pipes   were   to   be  determined,    proceed  thus: 
Assume  a  trial  value  for  Q  and  find  from  diagram  the  friction- 
head  per  1000  ft.  length  of  pipe  of  given  diameter  d,  thence  the 
value  of  hp  for  actual  leng  h  of  EJ.    Values  of  hp'  and  hp" 
corresponding  to  hp  are  now  noted  and  corresponding  values  of 
Q'  and  Q"  found  from  the  diagram  for  respective  diameters  df 
and  d".     The  sum  Q'  +  Q"  should  be  equal  to  Q.     If  such  is  not 
the  case  as  a  result  of  the  first  trial,  assume  a  new  value  for  Q; 
and  so  on,  until  the  necessary  equality  is  obtained. 

In  the  above  it  is  supposed  that  water  flows  into  Rf  and  R" , 
and  out  of  R;  but  if  Rf  is  at  a  sufficient  elevation,  or  if  pipe  EJ 
is  small  in  diameter,  water  may  flow  out  of  Rf,  as  well  as  out 
of  R.  In  such  a  case  the  summit  C  would  be  lower  than  the 
surface  in  R',  and  Q  +  Q'  =  Q". 

Similar  principles  and  methods  apply  to  any  system  or 
network  of  pipes. 

124.  Numerical    Problems.     (IV)  Supply-pipe    for    Turbine. 
Loss  of  Head. — In  previous  problems  of  this  chapter  examples 
have  been  treated  in  which  the  water  reaches  the  atmosphere 
at  the  lower  level  without  having  given  up  energy  for  any 
useful  purpose,  some  or  all  of  its  energy  having  been  expended 
in  fluid  friction.     Let  us  now  consider  the  case  of  a  turbine 
supplied  with  water  through  a  supply-pipe  of  riveted  steel, 
2000  ft.  in  length.     See  Fig.  96.     The  suction-head  (for  the 
short  draft-tube)  is  10  ft.;   whole  head,  80  ft.     The  consump- 


§  12 


NUMERICAL    PROBLEMS.      PIPES. 


201 


tion  of  water  in  steady  flow  is  limited  to  20  cub.  ft  per  sec. 
How  much  of  the  total  head  of  80  ft.  will  be  lost  in  the  supply- 
pipe,  and  correspondingly  how  much  power  lost  in  fluid  fric- 
tion? 

Solution. — We  find  from  the  friction-head  diagram  (in 
Appendix)  that  a  flow  at  rate  of  20  cub.  ft.  per  sec.  in  a  pipe  of 
24  in.  diameter  implies  a  mean  velocity,  v,  of  6.4  ft.  per  sec.; 
and  also,  if  the  pipe  is  of  clean  cast  iron,  a  friction-head  of 
5.8  ft.  per  1000  ft.  length;  that  is,  of  11.6  ft.  for  2000  ft.  length. 
Multiplying  this  11.6  by  |f£  for  riveted  steel  pipe  (see  §  115), 
we  obtain  15.8  ft.  as  the  friction-head  from  E  to  K.  This 
15.8  ft.  is  the  "drop,"  FD,  in  the  hydraulic  grade-line,  while 
C7F,  =  (v2+  20)  (1  +0.5),  =1.02  ft.  Hence  the  open  piezometer 
height  DK,  at  K  (taking  CK  as  70  ft.),  is  70  -(15.8  +  1.02), 
=  53.18  ft.;  and  the  vertical  distance  from  summit  D  to  tail- 
surface  T  is  63.18  ft.  In  computing  the  efficiency  TJ,  of  the 
turbine,  (in  a  test,)  from  the  expression  y  =  R'v'  +  Qj>ht  we  should 


PIEZOMETER 


Q  =20   CUB.  FT.  PER  SEC. 


FIG.  96. 


write  for  h  the  value  63.18 +  (^-j- 20);  i.e.,  63.86  ft.j  and  not 
80  ft.;  since  the  24-in.  pipe  is  not  a  part  of  the  turbine,  Again, 
referring  to  Fig.  45,  the  hi  of  that  figure  would  be  represented 
by  DK  +  (v2+  2g),  i.e.,  by  53.86  ft.,  in  the  present  case;  and 
hn  by  -10  ft.;  that  is,  the  .h,  =hi-hn,  of  Fig.  45  will  be  (as 


202  HYDRAULIC    MOTORS.  §   124a. 

already  stated)  53.86-  (-10),  =63.86  ft.,  for  the  purposes  of 
the  present  problem. 

If,  then,  a  diameter  of  24  in.  be  adopted  for  the  2000  ft. 
supply-pipe,  the  loss  of  head  thereby  occasioned  is  about  16  ft. 
(  =  h*)  and  the  loss  of  power  is*  Qrh2,  =20X62.5X16,  =20,000 
ft.-lbs.  per  sec.  ;  or  36.4  H.P. 

As  the  loss  of  head  of  16  ft.,  in  the  supply-pipe  of  24  in. 
diameter,  is  about  one-fifth  of  the  total  head  (80  ft.)  of  the 
mill-site,  it  will  be  instructive  to  note  the  great  reduction  in 
this  loss  of  head  as  due  to  an  increase  in  the  diameter  of  the 
supply-pipe  from  2  ft.  to  3  ft.,  Q  remaining  as  before  (20  cub. 
ft.  per  sec.).  For  a  36-in.  pipe,  from  the  friction-head  diagram 
for  clean  pipes  we  find  /i2  =  0.73  ft.  for  1000  ft.  length,  and  hence 
(0.73X2  =  )  1.46  ft.  for  the  actual  2000  ft.  length.  If  1.46  be 
multiplied  by  |ff,  as  before  (for  riveted  steel  pipe),  the  result 
is  a  loss  of  head  of  only  2  ft.;  instead  of  the  16  ft.  when  the 
diameter  was  24  inches.  However,  in  an  actual  case  in  prac- 
tice, the  annual  interest  on  the  extra  cost  of  the  36-in.  pipe 
might  be  greater  than  the  annual  income  from  sale  of  power 
due  to  the  head  so  saved  (14  ft.).  Commerical  considerations 
of  this  nature  are  of  great  importance  in  situations  where  long 
supply-pipes  are  needed  to  develop  a  water-power. 

i24a.  Power  Lost  in  a  Supply-pipe.  —  In  general,  in  this  con- 
nection, it  is  to  be  noted  that  if  in  the  expression  for  the  friction- 

4fl  v2 

head  in  a  long  pipe  [eq.  (1),  §  115],  viz.,  ^F  =  ~J"^~  ,  there  be 

a    ^g 

/    s!2\ 

substituted  for  v  its  equivalent  Q+l  —r-  )  ,  we  have 

,       32/Z  Q2 


from  which  it  is  seen  that  if  the  coefficient  /  be  considered  con- 
stant (as  a  rough  approximation),  the  friction-head  is  inversely 
proportional  to  the  fifth  power  of  the  diameter  d,  for  a  con- 
stant Q.  Evidently,  then,  an  increase  in  the  diameter  produces 
a  relatively  large  decrease  in  the  friction-head,  as  has  just 
been  illustrated. 


§  125.  WATER-HAMMER   IN    PIPES.  203 

Again,  as  to  the  power  lost  in  a  supply-pipe,  Lp  ft.-lbs. 
per  sec.,  we  have 


on  which  the  statement  may  be  based,  as  approximately  true, 
that  the  power  lost  in  a  supply-pipe  is  directly  proportional 
to  the  cube  of  the  volume  of  flow  (Q  cub.  ft.  per  sec.)  and  in- 
versely to  the  fifth  power  of  the  diameter  (d)  of  pipe.  For 
instance,  doubling  the  discharge,  without  change  in  length  or 
diameter,  would  involve  about  eight  times  as  much  loss  of  power 
in  the  supply-pipe. 

i24b.  Note.—  If  M,  in  Fig.  96,  were  a  centrifugal  pump 
(instead  of  a  turbine)  requiring  a  power  Pi/  to  drive  it,  pumping 
20  cub.  ft.  of  water  per  sec.  from  T  to  A,  the  summit  D'  of  the 
piezometer  column  at  K  would  stand  at  a  height  DfK  above 
K  equal  to  ~CK  +  hF;  or  for  a  24-in.  pipe  70  +  15.8=85.8  ft.; 
and  therefore  15.8  ft.  above  C.  See  §§  12  and  13.  The  hy- 
draulic grade-line  would  then  be  a  straight  line  from  D'  to  a 
point  in  A  vertically  above  E. 

125.  Water-hammer  in  Pipes.  Unsteady  Flow.  —  When  the 
water  supplying  a  turbine  is  conducted  through  a  very  long 
pipe,  flowing  with  some  velocity  v,  a  more  or  less  sudden 
closing  of  the  wheel-gates  may  cause  high  bursting  pressures 
within  the  pipe,  unless  relief-valves  are  provided,  or  a  stand- 
pipe  communicating  with  the  supply-pipe  just  up-stream  from 
the  wheel-gates.  Without  such  provision  the  arresting  of  the 
motion  of  the  large  mass  of  water  in  the  pipe  creates  a  great 
increase  of  pressure  of  the  water  against  the  walls  of  the  pipe, 
sufficient  in  some  cases  to  rupture  it.  The  most  extreme 
instance  of  this  kind  would  be  occasioned  by  the  instantaneous 
closing  of  a  valve-gate  in  a  pipe  in  which  water  is  flowing. 
This  will  now  be  investigated.  If  the  pipe  does  not  move 
lengthwise,  the  original  kinetic  energy  of  the  water  will  ex- 
haust itself  in  compressing  the  water  itself  and  in  distending 
the  walls  of  the  pipe.  In  our  first  treatment  the  walls  of  the 
pipe  will  be  considered  as  inextensible;  that  is,  their  disten- 


204  HYDRAULIC    MOTORS.  §  126. 

sion  will  be  neglected.  The  maximum  (unit)  fluid  pressure 
to  be  determined,  as  due  to  the  arrest  of  the  motion,  will  be 
that  over  and  above  the  pressure  already  existing  before  the 
interruption  of  the  condition  of  steady  flow,  and  may  be  called 
the  "  excess-pressure." 

126.  Water-hammer  in  a  Pipe.  Distension  of  Pipe  Neg- 
lected.— We  shall  at  first  disregard  the  distension  of  the  pipe 
walls  due  to  increase  of  internal  pressure.  As  regards  the  com- 
pressibility of  water  it  is  known  from  physics  that  water  has 
only  one  kind  of  modulus  of  elasticity,  viz.,  that  of  change  of 
volume  (or  " Bulk-modulus"),  which  may  be  called  E.  If  a 
mass  of  water,  of  original  volume  V,  is  by  compression  from  all 
sides  reduced  in  volume  by  an  amount  ^F,  the  fluid  pressure 
so  far  induced  being  p  Ibs.  per  sq.  in.,  then  E  is  defined  as  the 
quotient  p  +  relative  change  of  volume,  i.e., 

F         P         PV 

h==W^v=7v'    •••••• 

For  pressures  below  p  =  1000  Ibs.  per  sq.  in.  (and  at  ordinary 
temperatures)  E  may  be  taken  as  294,000  Ibs.  per  sq.  in.  (For 
very  high  pressures,  see  Engineering  News,  Oct.  4th,  1900,  p. 
236.) 

In  Fig.  97  we  have  a  horizontal  pipe  of  indefinite  extent  in 
which  at  first  water  is  flowing  (from  left  to  right)  with  a  con- 
stant velocity  of  c  ft.  per  second,  the  valve-gate  G  being  open. 
The  pipe  is  non-distensible.  If  now  the  gate  G  is  instantaneously 
closed,  passing  into  position  GCf ,  the  vertical  laminae  of  water 
on  the  left  of  the  gate  crowd  up  against  it,  and  at  the  end  of 
a  short  time,  dt  seconds,  all  the  laminae  up  to  some  position 
BB',  a  distance  ds'  from  (7,  have  come  to  rest,  with  reduced 
volume  and  under  some  pressure  p  (excess  pressure)  whose 
value  we  wish  to  determine.  At  the  beginning  of  this  short 
time  dt  there  were  certain  laminae  in  the  position  A  A' 
which  at  the  end  of  the  time  dt  have  just  reached  position  BB', 
having  traveled  a  distance  AB,  =ds,  without  reduction  of 
volume  and  with  unchecked  velocity  c;  so  that  c  =  ds+dt. 
That  is,  a  "wave  of  compression"  travels  from  C  to  B  in  time 


§126. 


WATER-HAMMER   IN    PIPES. 


205 


dt,  and  hence  the  velocity  of  the  "wave  front,"  or  of  "wave 
propagation/'  is  ds'  +  dtj  which  may  be  called  C,  or  the  velocity 
of  sound  in  water. 

Therefore,  in  a  time  dt  the  prism  of  water  AA'C'C,  whose 
original  volume  was  V  =  F(ds+ds'),  (where  F  is  the  sectional 
area  of  the  pipe,)  has  had  its  velocity  changed  (different  laminae 


p 

PE 

G 

r*S 

G/ 

ITE 

\ 

? 

\Y::-# 

^fl 

«•  . 

"V.- 

•%•  » 

Ov^: 

•'»'::•  •'/>' 
V/.X:-.O: 
^M:"V-:V.': 

•    -      »     ^. 

•!«••  'V'-' 

*  ^«  B.  \ 

;  '•>-".  ,-  •-•*.  .v 

••c    .v. 

::"::  •'$?$. 

•'.' 

*  •  V  ':  '  .' 

-V-';-'.- 

%$$* 

- 

::  :A' 

\Y-B'$$$ 

."'•'",-• 

V-';'"-.Y 

iS^'c' 

- 

•- 

-di" 

FIG.  97. 


successively)  from  c  to  zero  and  has  undergone  a  change  of 
volume  of  JV  =  Fds.  Each  of  the  vertical  laminaB  composing 
this  prism  has  encountered  a  retarding  force  increasing  regularly 
from  zero  up  to  its  final  maximum,  P  =  pF,  and  we  may  for 
simplicity  assume  that  the  value  of  this  final  maximum  pressure 
is  the  same  as  if  the  prism  in  question  had  remained  rigid;  that 
is,  had  remained  of  unaltered  length  AC  while  describing 
the  distance  ds  in  being  brought  to  rest;  its  retardation  being 
brought  about  by  an  imponderable  spring  (say) ,  the  compressive 
force  in  which  increases  progressively  in  proportion  to  the 
amount  of  shortening  of  the  spring,  from  zero  to  P. 

Now  for  a  uniformly  retarded  motion  we  have  from  eq.  (3), 
p.  54,  of  M.  of  E.,  when  the  initial  velocity  is  c  and  the  final 
is  zero,  O2  — c2  =2  X  distance  X  acceleration.  The  motion  of  the 
prism  in  the  present  case  is  not  uniformly  retarded;  that  is, 
the  (negative)  acceleration  is  not  constant;  but  we  may  use 
the  relation  just  quoted  if  we  substitute  the  average  accelera- 
tion, which  is  one-half  of  its  final  value,  viz.,  —  J(pF-f- 


206  HYDRAULIC   MOTORS.  §  127. 

=  —$pF+  [F(ds+ds')r+  g].     The  result  of  such  substitution  is 
c2  =  pg-ds+(ds+ds')r',  ....  (8a) 

but  since  c  =  ds-r-  dt,  this  may  be  written 

p-g-dt=(ds  +  ds')rc  .......     (9) 

Also,  from  definition  of  E  (see  eq.  (8)), 

or  E= 


ds 


Dividing  (9)  by  (10)  we  have  P^  =  ~-,    ....    (11) 

i/ 

or  P=c>      .......     (12) 


for  the  value  of  the  "  excess  pressure."  It  is  seen  to  be  pro- 
portional to  the  original  velocity,  c,  of  the  water  in  the  pipe. 

Incidentally,  we  may  now  determine  the  velocity  of  sound 
in  water,  (7;  viz.,  by  multiplying  eq.  (9)  by  (10),  whence 

Egds-dt=(ds+ds')2rc  ......     (13) 

Now  ds  is  usually  so  small  compared  with  ds'  that  we  may 
neglect  it  when  added  to  the  latter,  and  thus  obtain  Egds-dt  = 
(ds')2rc.  But  ds  +  dt=c,  and  ds'+  dt  =  C;  therefore,  finally, 


C=^ (14) 

With  E=  294,000  Ibs.  per  sq.  in.,  g=  32.2  (ft.  and  sec.),  and 
f  =  62.5  Ibs.  per  cub.  ft.,  this  gives  C  =  4670  ft.  per  sec. 

Eq.  (12)  may  be  written  in  this  form  (taking  #=294,000 
Ibs.  per  sq.  in.  and  ?-=62.5  Ibs.  per  cub.  ft.): 

p  (in  Ibs.  per  sq.  in.)  =  63  X[c  in  ft.  per  sec.].     .     (14a) 

127.  Water-hammer  in  a  Pipe,  Distension  of  Pipe  Considered. 
(See  Fig.  98.) — In  this  case,  the  water  in  the  pipe  being  originally 
in  motion  in  steady  flow  from  left  to  right  with  velocity  c,  let 
the  gate  G  be  suddenly  closed,  into  position  GH' ';  and  let 
BB'  be  the  position  of  the  "  wave  front "  at  the  end  of  dt  seconds 


§127. 


WATER-HAMMER   IN   PIPES. 


207 


after  the  closure.  The  compressed  prism  of  water,  which 
originally  occupied  the  position  and  space  AA'C'C,  its  volume 
being  then  V  =  F(ds+ds'),  is  now  found  to  occupy  the  space 
BB'H'H  (dotted  sides),  the  pipe  having  been  distended,  and 
its  radius  having  increased  from  a  value  r  to  a  new  value, 


B 


H 


PIPE 


* 


H 


FIG.  98. 


r+Jr,  (see  the  end-view  on  the  right,  where  the  thick  outline 
shows  the  original  size  of  the  pipe.)  The  change  (decrease)  of 
volume  of  this  prism  is  evidently  AV  =  F  -ds-Zxr-Ar-ds' ',  where 
F  is  sectional  area  of  pipe,  =  nr2,  and  hence  [see  eq.  (8)], 


E= 


pF(ds+ds') 
Fds-2nr-4r-dsf' 


(15) 


By  the  same  reasoning  as  in  the  previous  paragraph  we  may 
repeat  eq.  (9),  viz., 


p-g-dt  = 


')  j-c. 


(16) 


The  unit  pressure  being  p  at  this  instant,  acting  also  as  a 
bursting  pressure  radially  outward  on  the  inner  surface  of  the 
pipe-wall,  between  B  and  H,  the  simultaneous  tensile  stress  (or 
*'  hoop-tension  ")  in  the  pipe-wall,  p'  Ibs.  per  sq.  in.,  will  have 
a  value  of  p'  =  rp  +  tf,  where  tf  is  the  thickness  of  the  pipe- wall 
Isee  p.  537,  M.  of  E.,  eq.  (2)].  Now  if  E'  is  the  modulus  of 
elasticity  (linear;  Young's  modulus)  of  the  metal  of  which 
the  pipe  is  made,  and  X  is  the  increase  of  length  of  the  circum- 
ference of  the  pipe  due  to  stress  p',  we  have  (see  p.  203,  M. 


208  HYDRAULIC    MOTORS.  §  128. 

of  E.),  by  definition, 


or         ^ 

But,  from  proportion,  A:2?rr:  :Jr:r,  or  A  =  27r-Jr;  hence 

pr2 

*~FF  ........ 

If  this  be  substituted  in  (15)  and  F  replaced  by  xr2,  we 
finally  obtain  [see  also  (16)],  the  relation 

^i_9.  ^  JL-&. 
dt         '  dt't'E'~Ecf 

But  if  in  (16)  we  neglect  ds  when  added  to  ds'  ',  writing  C 
for  ds'  +  dt,  we  obtain 


(19) 


which  may  be  substituted  in  (18)  and  a  solution  made  for  C 
(note  being  made  that  ds+dt  =  c  and  that  dsf  +dt  =  C),  whence 


._J£~~P: 

>r  (t'E'+l 


(20) 


as  the  (diminished)  velocity  of  sound  *  along  the  water  in  the 
pipe  now  that  the  distension  of  the  latter  is  brought  into  play; 
and  therefore  [see  (19)] 


=C\N( 


g(t'E'+2rE)} 


is  the  "excess  pressure  "  tending  to  burst  the  pipe. 

(N.B.  These  same  results  could  also  be  obtained  by  putting 
the  original  kinetic  energy  of  the  prism  AA'C'C  equal  to  the  work 
of  compressing  itself  and  of  distending  the  pipe-  wall;  see  §  196, 
M.  of  E.) 

128.  Joukovsky's  Experiments  on  Water-hammer.  —  That 
formulae  (20)  and  (21)  are  practically  true  has  been  demon- 
strated by  Prof.  Joukovsky  in  experiments  conducted  at 

*  First  proved  by  Korteweg  in  1878.  See  also  Mr.  J.  P.  FrizelTs  book 
on  "Water  Power,"  New  York,  J.  Wiley  <fe  Sons,  1901. 


§  128.  WATER-HAMMER  IN    PIPES.  209 

Moscow,  Russia,  in  1897-98.  These  experiments*  were  made 
with  horizontal  pipes  of  cast  iron  of  four  different  lengths, 
viz.,  2494,  1050,  1066,  and  7007  ft.;  their  diameters  being  2, 
4,  6,  and  24  inches,  respectively. 

It  was  found  that  so  long  as  the  time  of  closing  the  valve 
was  less  than  that  required  for  the  wave  of  compression,  or 
sound  wave,  to  make  a  "round  trip"  from  the  valve  to  the 
reservoir  from  which  the  pipe  issued  and  back  to  the  valve, 
the  effect  was  practically  the  same  as  if  the  closure  had  been 
instantaneous.  The  wave  being  reflected  down  the  pipe  from 
the  water  in  the  reservoir,  the  time  for  the  "  round  trip/'  if 
I  denote  the  length  of  the  pipe,  is  tr  =  2l/C.  It  was  found 
that  when  the  time,  £",  of  closure  was  longer  than  tr,  the  excess 
pressure  produced,  p",  was  less,  and  in  the  same  proportion 
as  tr  was  less  than  t" \  that  is,  that  p" :p:  :tr:t". 

On  account  of  the  elasticity  of  the  water  its  condition  of 
compression  is  only  temporary,  being  followed,  during  the 
"recoil,"  as  it  may  be  called,  by  a  period  of  "rarefaction  "  or 
of  pressure  below  the  original  or  normal  pressure;  thus  there 
occur  at  the  gate  successive  pulsations  of  pressure  a  complete 
cycle  of  which  is  equal  to  the  time  of  two  "  round  trips. "  These 
pulsations  of  pressure  diminish  gradually  in  intensity  through 
friction. 

In  the  case  of  a  pipe  of  smaller  diameter  connected  with  the 
main  pipe  and  terminating  in  a  "dead  end"  or  valve  per- 
manently closed,  a  much  greater  excess  pressure  is  produced 
in  the  smaller  pipe — about  double  that  in  the  main  pipe. 

Some  practical  conclusions  reached  as  the  result  of  these 
experiments  are  quoted  (see  foot-note  below):  "The  simp- 
lest method  of  protecting  water-pipes  from  water-hammer 
is  found  in  the  use  of  slow-closing  gates.  The  duration  of 
closure  should  be  proportional  to  the  length  of  the  pipe-line. 
Air-chambers  of  adequate  size  placed  near  the  valves  and 
gates  eliminate  almost  entirely  the  hydraulic  shock,  and  do  not 
allow  the  pressure  wave  to  pass  through  them;  but  they  must 

*  A  good  resume  of  these  experiments  was  published  in  the  Proceedings 
for  1904  of  the  Amer.  Water-works  Assoc.,  p.  335. 

'4 


210  HYDRAULIC   MOTORS.  §  129. 

be  very  large  and  it  is  difficult  to  keep  them  supplied  with  air. 
Safety-valves  allow  to  pass  through  them  pressure  waves  of 
only  such  intensity  as  corresponds  to  the  elasticity  of  the  springs 
of  the  safety-valves/7 

129.  Time  of  Closure  Longer  than  tr. — When  the  time  of 
closure  is  very  much  longer  than  that,  tr,  for  the  "  round  trip/' 
the  rate  at  which  the  opening  of  the  valve-gate  is  closed  up 
would  seem  to  have  an  important  bearing  on  the  rise  of  pres- 
sure produced.     Theoretical  investigations  along  this  line  have 
been  made  by  Mr.  B.  F.  Latting,  C.E.,  and  the  present  writer; 
and  a  few  experiments  were  also  made  by  Mr.  Latting,  the 
results  of  which  were  fairly  confirmatory  of  theory.     See  the 
Engineering  Record  for  Feb.  25,  1905,  p.  214,  or  Engineering 
of  March  17,  1905,  p.  363;  also  Transac.  Assoc.  Civ.  Engineers 
of  Cornell  University,  for  1898,  p.  31. 

130.  Water-hammer.     Numerical  Examples. — (I)  If  the  orig- 
inal velocity  of   the   water  in   a   2-in.  pipe   is   4  ft.  per  sec. 
and  a  valve-gate  is  closed  instantaneously,  what  excess  pressure 
is  produced? 

This  pipe  being  small  in  diameter,  eq.  (14a)  may  be  used, 
from  which  we  have  p  =  63X4  =  252  Ibs.  per  sq.  inch. 

If  the  length  of  the  2-in.  pipe  is  1000  ft.  the  same  pres- 
sure would  be  produced  so  long  as  the  time,  t',  of  closing  the 
valve  was  less  than  tr  =  2x1000  -4670,  =0.428  sec.  If  the 
time  of  closing  were  longer  than  0.428  sec.,  the  excess 
pressure  (p")  would  be  less  in  accordance,  with  the  relation* 

P"=(tr+t")p. 

If  the  2-in.  pipe  were  only  200  ft.  long  the  full  water-hammer 
of  p  =  63x4,  or  252  Ibs.  per  sq.  in.,  would  not  be  produced, 
unless  the  time  t"  were  less  than  0.085  sec. 

(II)  A  riveted  steel  pipe  is  5  ft.  in  diameter,  the  thickness 
of  pipe-wall  being  \  inch.  The  water  within  it  has  origi- 
nally a  velocity  of  4  ft.  per  sec.  What  is  the  full  excess 
pressure  of  water-hammer  if  Ef  be  taken  as  30,000,000  Ibs.  per 
sq.  in.? 

We  now  substitute  in  eq.  (21)  and  obtain  p=137  Ibs.  per 
sq.  in.  Also  from  eq.  (19)  we  have  for  the  velocity  of  the 


§   131.  WATER-HAMMER   IN    PIPES.  211 

compression  wave 

„    137X144X32.2 

°-         4X62.5        =2552  ft.  per  sec. 

In  case  the  length  of  the  pipe  is  7000  ft.  the  full  value  of  p, 
=  137  Ibs.  per  sq.  in.,  would  not  be  produced  unless  the  time 
of  closing  were  less  than  tr,  which  =  2x7000 -^2552  =  5.48  sec.; 
and  similarly  for  other  values  of  the  length. 

The  "  hoop-tension  "  in  the  wall  of  the  pipe,  due  to  the  excess 
pressure  p,  would  be  p"  =  rp+  (thickness),  i.e. 

p  =  30  X 137  -i  =  16,440  Ibs.  per  sq.  in. 

To  this  would  have  to  be  added  the  hoop-stress  due  to  original 
fluid  pressure;  and  the  weakening  of  plates  due  to  riveting 
would  have  to  be  considered.  Evidently  the  total  hoop-stress 
would  be  too  great  for  safety. 

131.  Prevention  of  Water-hammer  with  Turbines. — The 
prevention  of  much  increase  of  pressure  at  the  turbine  end  of  a 
long  penstock  is  not  only  desirable  for  the  safety  of  the  pen- 
stock itself,  but  also  in  some  cases  absolutely  necessary  for  the 
proper  regulation  of  the  motor. 

For  instance,  when  the  resistance  or  "load"  on  the  turbine 
diminishes,  and  when  consequently  by  the  action  of  the  govern- 
ing apparatus  the  wheel-gates  begin  to  close,  in  order  that  by  the 
diminution  of  the  rate  Q  (cub.  ft.  per  sec.)  of  water  used  by 
the  wheel  the  working  force  exerted  on  the  wheel  may  be  re- 
duced, so  great  a  rise  of  pressure  might  be  produced  just  outside 
the  gates  as  to  bring  about  an  increase,  instead  of  a  decrease,  in 
the  working  force  acting  on  the  wheel;  and  thus  produce  an 
effect  just  the  contrary  of  that  intended.  Provision  therefore 
is  often  made  for  the  escape  of  some  of  the  water  through  a 
side  outlet  or  " by-pass"  leading  to  the  atmosphere;  which  is 
only  opened,  and  that  automatically,  whenever  the  pressure 
increases  slightly  above  its  normal  value.  The  valve  closing 
this  outlet  is  called  a  "relief  valve."  (See  p.  422  of  the  Engi- 
neering News  of  Nov.  1904,  where  a  valve  disc  23  in.  in  diameter 


212  HYDRAULIC   MOTORS.  §  131. 

is  described,  with  its  appurtenances;  made  by  the  Lombard 
Governor  Co.). 

Another  method  of  preventing  any  material  increase  of 
pressure  in  the  penstock  when  the  turbine  gate  is  being  lowered 
is  by  the  use  of  a  stand-pipe  of  large  diameter  communicating 
with  a  side  opening  in  the  penstock  near  the  wheel.  When  the 
consumption  of  water  is  normal  and  the  flow  steady  the  water 
in  this  pipe  is  at  rest  and  stands  at  a  height  reaching  to  the 
hydraulic  grade-line  (see  DK  in  Fig.  96) .  When  the  wheel-gate 
closes  more  or  less,  a  part  of  the  flow  from  the  penstock  passes 
into  the  stand-pipe  and  spills  over  its  upper  edge;  and  the  rise 
of  pressure  near  the  wheel-gate  is  not  excessive.  Conversely, 
when  the  wheel-gates  open  beyond  the  normal  position  the 
extra  flow  desired  is  at  first  furnished  by  the  water  in  the  stand- 
pipe  and  the  pressure  just  above  the  wheel-gates  does  not  fall 
to  too  low  a  value  while  the  water  in  the  penstock  is  adjusting 
itself  to  a  new  and  greater  velocity  of  steady  flow.  In  Fig. 
99  is  shown  the  terminal  arrangement  of  a  long  penstock  in 
Fall  Creek  gorge  at  Ithaca,  N.  Y.  This  penstock,  of  some  6  ft. 
diameter  and  about  1000  ft.  long,  supplied  two  pair  of  30-in. 
"  New  American"  turbines  on  horizontal  shafts  (see  also  Fig.  63), 
working  under  90  ft.  head,  with  draft-tubes  as  shown.  In  the 
upper  part  of  the  figure  is  seen  the  lower  part  (only)  of  a  stand- 
pipe  or  "  relief  -pipe  "  42  in.  in  diameter  and  47  ft.  high.  Two 
air-chambers  are  also  provided,  one  in  each  branch  of  the  pen- 
stock, just  above  each  wheel-case  (containing  a  pair  of  turbines, 
as  shown  in  the  figure). 

The  use  of  a  stand-pipe  is  considered  the  best  method  of 
obviating  water-hammer,  etc.,  in  the  case  of  a  turbine  supplied 
by  a  long  penstock  when  the  head  is  not  too  great  and  freezing 
can  be  prevented.  With  impulse- wheels  supplied  through  a  long 
penstock  the  rate  at  which  water  is  used  by  the  wheel  (Pelton, 
for  instance)  is  sometimes  varied  by  the  use  of  a  "deflecting- 
nozzle  "  through  whose  lateral  or  downward  movement,  con- 
trolled by  the  governor,  more  or  less  of  the  jet  passes  on  with- 
out acting  on  the  buckets.  In  the  Cassell  impulse-wheel  (see 
Engineering  News,  Dec.  1900,  p.  442)  the  two  lobes  or  halves  of 


FIG.  99. 


213 


214  HYDRAULIC    MOTORS.  §  132. 

each  bucket  are  caused  to  separate  more  or  less  by  the  action  of 
the  governor,  and  the  same  object  is  thus  accomplished ;  a  por- 
tion of  the  jet  passing  between  the  two  parts  of  the  bucket  and 
without  action  on  it.  In  this  way  there  is  no  checking  of  the 
velocity  of  the  water  in  the  supply-pipe  and  water-hammer 
is  completely  avoided;  but  of  course  such  a  device  is  not 
economical  of  water. 

132.  Open  Channels,  or  Canals. — Since  these  are  frequently 
used  to  conduct  water  from  a  reservoir  to  a  wheel-pit  or  to 
the  inlet  of  a  pipe  or  penstock,  for  supplying  a  hydraulic  motor, 
a  few  pages  will  be  given  to  their  consideration  in  the  present 
work  in  addition  to  what  is  already  presented  in  the  author's 
Mechanics  of  Engineering. 

The  situation  usually  presented  is  that  of  " uniform  motion" 
in  steady  flow.  By  this  it  is  implied  that  the  body  of  water  in 
motion  is  of  indefinite  length  and  has  the  form  of  a  geometric 
prism,  i.e.,  the  surfaces  of  the  bed,  banks,  and  of  the  water 
itself  are  parallel,*  the  mean  velocity  of  the  water  in  any  section 
is  equal  to  that  in  any  other  and  does  not  change  with  lapse 
of  time  (see  p.  756,  M.  of  E.).  The  flow  will  not  be  of  this 
character,  however,  unless  the  quantities  concerned  bear  a 
certain  relation  to  each  other.  These  quantities  (as  concerned 
in  the  most  widely  used  formula,  Kutter's  Formula,  for  uniform 

motion)  are  the  ratio  called  the  "slope"  S=T,  where  h  is  the  fall 

of  the  surface  (and  also  that  of  the  bed)  in  a  length  I  along  the 
channel;  the  "  hydraulic  radius, "  or  "  hydraulic  mean  depth,"  R, 
=  area  of  cross-section,  F,  divided  by  the  wetted  perimeter; 
the  mean  velocity,  v,  of  flow  (about  0.83  of  the  surface-velocity 
in  mid-stream);  and  a  "coefficient  of  roughness,"  n,  dependent 
on  the  character  of  the  surface  of  bed  and  banks.  For  uniform 
motion,  then,  to  subsist,  the  relation  which  must  hold  between 
these  quantities,  as  expressed  in  Kutter's  Formula  (which  is 
fairly  well  supported  by  a  wide  range  of  experiments;  though 
considerable  uncertainty  must  generally  prevail  in  matters  of 
this  kind),  is  (for  the  English  foot  and  second  as  units) 

*  That  is,  parallel  to  an  axis. 


§  132.        KUTTER'S  FORMULA  FOR  UNIFORM  MOTION.  215 

L8U   aoo 

n  s  .  _ 

VRs'<    •    • 


or,  for  brevity,  v  =  A\/Rs,    .......     (2) 

where  A  stands  for  "  Kutter's  coefficient"  in  the  bracket  in  (1). 

The  ordinary  scheme  of  values  for  n  is  here  appended,  viz.  : 
n  =  .009    for  well-planed  timber  evenly  laid. 

.010;  plaster  in  pure  cement;  glazed  surfaces  in  good  order. 
.011;  plaster  in  cement   with  one-third    sand;    iron    and 

cement  pipes  in  good  order  and  well  laid. 
.012;  un  planed  timber,  evenly  laid  and  continuous. 
.013;  ashlar  masonry  and  well  laid  brickwork;    also  the 
above  categories  when  not  in  good  condition  nor 
well  laid. 

.015;  "  canvas  lining  on  frames";   brickwork  of  rough  sur- 

face; foul  iron  pipes;  badly  jointed  cement  pipes. 

.017;  rubble  in  plaster  or  cement  in  good  order;    inferior 

brickwork;  tuberculated  iron  pipes;  very  fine  and 

rammed  gravel. 

.020;  canals  in  very  firm  gravel;    rubble  in  inferior  condi- 

tion; earth  of  even  surface. 
.025;  canals  and  rivers  in  perfect  order  and  regimen  and 

perfectly  free  from  stones  and  weeds. 
.030  ;  canals  and  rivers  in  earth  in  moderately  good  order  and 

regimen,  having  stones  and  weeds  occasionally. 
.035  ;  canals  and  rivers  in  bad  order  and  regimen,  overgrown 
with    vegetation,    and    strewn    with    stones    and 
detritus. 

The  value  of  the  coefficient  A  is  most  readily  found  from  the 
diagrams  in  the  Appendix  of  this  book.  A  separate  diagram 
has  been  constructed  for  each  of  the  above  values  of  n  (and  also 
for  n  =  .040)  .  For  example,  for  a  hydraulic  radius  of  2  ft.  and 
a  slope  of  s  =  0.0002  which  is  0.2  ft.  per  thousand  we  find  that 
when  ft  =  .012  (unplaned  timber)  A  is  equal  to  139;  to  be  used 
with  the  English  foot  and  second  in  eq.  (2)  .  The  value  for  A 


216  HYDRAULIC    MOTORS.  §  132a. 

for  a  slope  of  10  ft.  per  thousand  will  also  hold  for  all  higher 
slopes  with  sufficient  accuracy  (as  is  also  evident  from  the 
diagrams)  .* 

The  student  should  guard  against  the  error  of  supposing  that 
eq.  (1)  or  (2)  would  hold  for  measurements  made  at  a  single 
cross-section  of  a  body  of  water  flowing  with  steady  flow  in  an 
open  channel.  The  depth,  area,  and  shape  of  cross-section, 
and  character  of  surface,  etc.,  must  be  the  same,  respectively, 
at  all  sections  of  a  fairly  long  reach  of  the  channel,  to  constitute 
a  case  of  uniform  motion  to  which  eq.  (1)  and  (2)  apply.  Prob- 
lems involving  non-uniform,  or  variable,  motion  (with  steady 
flow)  where  the  surface  is  not  parallel  to  the  bed  (in  longitudinal 
profile)  will  be  considered  later. 

i32a.  Coefficient  of  Fluid  Friction  for  Open  Channels.  —  If 
we  go  back  to  the  theoretical  basis  of  the  form  of  the  relation  in 
eq.  (2)  (see  pp.  757  and  758,  M.  of  E.),  we  find  the  formula  for 
uniform  motion  to  be 


§ 

•-S/T: 


(3) 


involving  /,  the  "coefficient  of  fluid  friction,"  corresponding  to 
that  for  flow  in  pipes.  In  other  words,  Kutter's  coefficient,  A, 
may  be  written  as  A=\/2g+f,  or 


Of  course,  while  /  is  an  abstract  number,  the  same  in  value 
whatever  units  of  measurement  and  time  are  selected,  A  is  not. 
Since  problems  are  to  be  treated  in  which  the  flow  is  not  "  uni- 
form" (although  "  steady"),  we  shall  need  the  quantity  /;  and 
this  may  conveniently  be  found  by  first  finding  A  from  a  diagram, 
as  if  the  case  were  one  of  uniform  motion,  and  then  determining 
/  from  eq.  (4)  .  Or,  vice  versa,  if  preferable,  we  may  replace  the 
/  of  a  formula  applying  to  a  non-uniform  steady  flow  (depths 
different  along  the  length  at  different  points,  e.g.)  by  its 

*  A  book  of  Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels; 
Uniform  Motion,  by  the  present  writer  (New  York,  J.  Wiley  &  Sons,  1902), 
obviates  the  necessity  of  numerical  substitution  in  the  use  of  Kutter's  for- 
mula (eq.  (1)  above)  f^r  all  practical  purposes. 


§133. 


FORMS    OF    SECTION.       OPEN   CHANNELS. 


217 


equivalent  in  terms  of  A;  thus  f=2g+A2;  but  if  values  of  A 
are  used  from  the  diagrams  (Appendix),  the  foot  and  second 
must  be  used  throughout  the  whole  formula  in  which  A  appears. 

133.  Forms  of  Section.  Open  Channels. — The  forms  of 
section  most  generally  employed  for  open  channels,  for  water- 
power,  or  for  irrigation  are  the  semicircle,  or  other  segment 
of  a  circle;  the  rectangle;  and  the  trapezoid  with  horizontal 
base;  occasionally  the  horseshoe,  if  the  channel  is  roofed  over 
or  is  in  tunnel. 

For  the  semicircular  section  running  full,  or  for  the  lower 
half  of  any  regular  polygon,  also  running  full,  the  hydraulic 
radius  R  is  equal  to  half  the  radius  of  the  inscribed  circle.  It 
is  also  worth  noting  that  any  such  half  regular  polygon  has  a 
minimum  wetted  perimeter  for  a  given  area  and  consequently 
is  of  the  most  advantageous  form  from  a  theoretical  point  of 
view;  i.e.,  to  deliver  a  maximum  quantity  of  water  per  sec., 
Q,  for  a  given  slope  of  bed,  given  area  of  water  prism,  and 
given  number  of  sides  for  the  polygon. 

It  is  also  to  be  noted  that  of  all  trapezoidal  sections  running 
full  and  having  a  common  side  slope,  or  angle  0,  (see  Fig.  100,) 
of  the  bank,  that  one  is  of  the  most  advantageous  form  whose 
three  sides  forming  the  wetted  perimeter  are  tangent  to  the 
semicircle  having  a  radius  CE  equal  to  the  depth  h  and  with 
its  diameter  in  the  surface  of  the  water;  and  its  hydraulic 
radius,  R,  is  equal  to  the  half-depth. 

According  to  Prof.  Bovey  (Hydraulics,  2nd  ed.,  p.  231)  the 
angle  ft  should  not  be  made  greater  than  the  values  given  below 
for  different  characters  of 
bank,  respectively: 
with  retaining  walls  63°  36 
with  stiff  earthen 

sides,  faced 45° 

with   stiff  earthen 

sides,  unfaced.  ..33°  41' 
with  sides  in  light  FIG  m 

or  sandy  soil..  ..26°  34' 
To  avoid  erosion  velocities  in  some  soils  may  have  to  be  limited 


218 


HYDRAULIC    MOTORS. 


134. 


to  2  ft.  per  second  or  under;  with  timber  or  rock  for  bottom 
and  sides,  however,  v  may  be  allowed  to  reach  values  of  6 
to  10  ft.  per  second. 

134.  Numerical  Example.  Open  Channel  Supplying  Wheel- 
pit. — An  open  channel  with  bottom  and  sides  of  "  average 
rubble  "  masonry  and  whose  depth  h  is  to  be  one-half  of  its 
width  is  to  conduct  a  water-supply  of  Q  =  120  cub.  ft.  per  sec. 
with  "  uniform  motion/'  with  a  fall  of  only  3  ft.  in  its  length 
of  1200  ft.  Compute  a  proper  value  for  the  depth  h.  See 
Fig.  101.  A  is  the  reservoir  and  C  the  wheel-pit. 

Solution. — The  sectional  area  being  2h2  and  the  wetted 
perimeter  4h,  the  hydraulic  radius  is  R=2h2+4Ji  =  h  +  2.  The 
coefficient  of  roughness,  n,  is  0.017  (for  " average  rubble") 


Q  £=120    CUB.  FT.  PER  SEC? 

FIG.  101. 

(see  §132);  and  the  slope  is  3'  -5- 1200'  =  0.0025,  i.e.,  2.5  ft. 
per  thousand;  but  as  the  value  of  R  is  as  yet  unknown,  it  is 
impossible  to  use  the  diagram  directly  for  finding  the  value 
of  Kutter's  coefficient  A. 

Since  the  values  of  A,  however,  range  between  65  and 
150  for  ordinary  cases,  a  value  of  ^.  =  100  may  be  assumed 
for  a  first  trial,  and  a  first  approximate  value  of  h  deter- 
mined, as  follows:  With  A  =  100  (for  the  foot  and  sec.)  and 
V  =  Q  +  F=  120-^2/z,2,  we  have  from  eq.  (2),  §  132, 

120  |/i  X  0.0025 

—  =  100\J o ,     or    ft5  =  288; 


2h2  ^         2 

i.e.,  h  =  3. 10  ft.,  as  a  first  approximation. 


§  135.  OPEN  CHANNELS.   EXAMPLE.  219 

The  corresponding  R  is  h +2  =  1.55'  for  which  and  the 
given  slope  of  2.5  per  thousand  we  find  in  the  diagram  for 
ft  =  0.017  a  value  of  94  for  A.  With  this  more  exact  value  of 
A  we  again  use  eq.  (2),  obtaining 

120  IhX  0.0025 

—  =  94^-          -,     or    A  =  3.19ft., 

as  a  second  approximation;  for  which  R  would  be  1.59  ft.  With 
this  new  R  and  the  given  slope  we  find  from  the  diagram  that 
A  does  not  differ  sensibly  from  94.  Hence  the  value  h  =  3. 19  ft. 
is  final. 

Owing  to  the  uncertainty  generally  involved  in  the  choice 
of  a  "  coefficient  of  roughness/'  n,  in  problems  of  this  class, 
results  obtained  must  be  looked  upon  as  only  approximate. 
They  may  be  as  much  as  five  per  cent,  in  error. 

(The  solution  of  this  problem  would  be  much  shortened  by 
the  use  of  the  diagrams  mentioned  in  the  foot-note  on  p.  216. 
These  diagrams  deal  with  v,  R,  and  s;  and  not  directly  with 
the  coefficient  A.) 

A  practical  matter  to  be  noted  in  the  problem  now  treated 
is  the  fact  that  where  the  water  passes  from  the  reservoir  A 
into  the  entrance  of  an  open  channel,  a  drop  of  the  surface 
will  occur  of  an  amount  equal  to  v2  +  2g;  which  in  the  present 
case  is  not  small. 

Since  v  =  120  +  2h2  =  5.86  ft.  per  sec.,  we  have  for  v2+2g, 
or  corresponding  velocity-head,  about  0.54  ft.;  (see  page 
Conversion  Scales,  in  the  Appendix).  This  drop  should  be 
allowed  for  in  arranging  the  position  of  the  bottom  of  the 
channel,  and  in  consequence  of  it  the  bottom  of  the  channel 
at  NO  should  be  placed  3'.2+0'.54  =  3.74  ft.  below,  the  sur- 
face'of  the  (still)  water  in  reservoir  A;  while  the  bottom  at 
B  should  be  3  ft.  lower  yet,  or  6.74  ft.  below  the  surface  of 
the  water  in  A. 

135.  Height  and  Amplitude  of  Backwater.* — If  an  obstruc- 
tion such  as  a  weir  or  dam,  for  water-power  purposes  or  otherwise, 
is  thrown  across  the  bed  of  a  stream  or  channel  of  indefinite 

*  See  Engineering  Record,  July  1892,  p.  91. 


220 


HYDRAULIC    MOTORS. 


§  136. 


length  and  of  regular  form,  in  which  originally  there  was  a 
" steady  flow"  with  " uniform  motion";  when  the  flow  again 
becomes  steady,  over  the  weir,  the  depth  of  water  just  above  the 
weir  is  greater  than  before  and  the  increase  of  depth  at  that 
point  is  called  the  height  of  backwater.  Also,  the  longitudinal 
profile  of  the  water  surface  above  the  weir  is  more  or  less  curved, 
the  depth  being  found  in  general  to  be  less  and  less  as  we  proceed 
up-stream.  The  greatest  distance  up-stream  from  the  weir  at 
which  any  increase  of  depth  is  perceptible  is  called  the  "ampli- 
tude of  backwater."  A  knowledge  of  this  distance  in  the  case  of 
a  proposed  weir  and  also  of  the  increase  of  depth  at  any  distance, 
occasioned  by  the  building  of  the  weir,  is  often  of  much  impor- 
tance ;  since  if  another  weir  were  built  up-stream  from  the  one 
proposed,  its  available  head  of  water  for  power  purposes  might 
be  affected  by  the  backwater  of  the  first. 

After  a  weir  has  been  built  and  a  steady  flow  resumed,  the 
conditions  of  flow  of  the  stream  below  the  weir  are  of  course 
unchanged. 

130.  Height  of  Backwater  for  a  Weir  not  Submerged.— 
Fig.  102  represents  a  vertical  section  of  an  overfall  weir  (see  pp. 

674  and  683,  M.  of  E.)  hav- 

TTLV?*^"  "  ing    a    sharp-edged    sill    or 

crest,  A,  higher  than  the 
surface  of  the  tail-water  or 
original  surface  of  the 
stream  and  with  its  up- 
stream face  vertical.  We 
suppose  that  the  whole  dis- 
charge Q,  cub.  ft.  per  sec., 
of  the  stream  is  passing 
over  the  weir  and  that  the 
air  has  free  access  under  the 
sheet;  and  that  there  are 
no  "end-contractions";  that 
is,  that  the  crest  terminates  in  two  vertical  faces  parallel 
to  the  axis  of  the  stream  (see  p.  686,  M.  of  E.)  forming  a 
"channel  of  approach."  These  conditions  justify  the  use  of 


FIG.  102. 


§  136.  WEIRS    AND    BACKWATER. 

Bazin's  formula  *  (p.  688,  M.  of  E.),  b  being  the  length  of  crest 
of  the  weir  (and  also  the  width  of  the  channel  of  approach)  and 
p  the  height  of  the  weir  above  the  horizontal  bottom  of  the 
channel  of  approach;  see  Fig.  102.  (The  stream  itself  may, 
however,  be  wider  than  the  weir.)  The  formula  is 

;....(!) 

in  which  /*'  has  the  value 

//  =  0.6075 +[0.0148-f- (h2  in  feet)].    ...     (2) 

Problem. — Required  the  height  p  of  weir  to  produce  a  given 
height  of  backwater  H,  b  and  Q  being  both  known,  as  also  d0, 
the  original  depth  of  the  stream  and  (still)  the  depth  of  the  water 
below  the  weir.  Evidently  we  have  do+H=y0  (see  Fig.  102) 
and  thus  y0  (the  total  new  depth  at  weir)  becomes  known.  For 
the  determination  of  p,  therefore,  we  have  eq.  (1)  above  and  the 
relation 

p+h2  =  yo (3) 

The  solution  is  best  effected  by  writing  (1)  in  this  form: 
0.55^  _          3Q 

1      '  y2  ^       ,7.     /7T~     I        *-> V*/ 


to  be  solved  by  trial  for  h2  (or  "  head  on  the  weir"). 

Example  — Let  the  channel  be  rectangular  in  section  with  a 
width  equal  to  that,  b,  of  the  weir  (which  is  of  the  form  just 
described  and  indicated  in  Fig.  102) ;  with  6  =  30  ft.  and  Q  =  310 
cub.  ft.  per  second;  while  the  original  depth  is  d0  =  3  ft.  It  is 
required  to  find  such  a  value  for  the  height  p  of  the  weir  as  to 
make  the  increase  of  depth  or  height  of  backwater,  H,  equal 
to  4.5  ft.;  or  the  total  depth  just  above  the  weir,  y0,  =  7.5  ft. 

First  assuming  /*2  =  3  ft.,  with  0.60  as  a  first  approximation 
for  /*',  the  right-hand  member  of  eq.  (4)  =0.619;  while  the  left- 
hand  member  becomes  equal  to 1.09. 

Trying  h2  =  2.5  ft.  with  //  still  equal  to  0.60,  we  find 

*  In  Bazin's  experiments  p  ranged  from  0.2  to  2  metres;    h2  from  0.05 
to  0.6  metres;  and  b  from  0.5  to  2  metres. 


222  HYDRAULIC    MOTORS.  §  137. 

the  right-hand  member  of  (4)  =  0.819,  and 
"    left-hand         "         "    "  =1.06 

Again,  with  h2  assumed  =  2.0  ft.,  and  hence  /*'  =  .6075+'  9     = 

0.6149,     the  right-hand  member  of  (4)  =  1.113 
"     left-hand         "         "    "  =  1.04 

We  may  therefore  conclude  without  further  trial  that  a  value 
of  /i2  =  2.1  ft.  will  serve  the  purpose.    Therefore  p  =  5.4ft. 

In  case  the  channel  of  approach  is  considerably  wider  than 
the  length  of  the  weir  crest,  6,  there  will  be  end-contractions 
and  we  may  use  the  formula  of  Francis  as  given  on  p.  687,  M. 
of  E. 

137.  Special  Forms  of  Weir.  Mr.  Rafter's  Experiments. — 
In  1899  experiments  were  made  at  the  Hydraulic  Laboratory 
of  the  College  of  Civil  Engineering  at  Cornell  University 
by  Mr.  G.  W.  Rafter  for  the  United  States  Board  of  Engineers 
on  Deep  Waterways  (see  vol.  xliv  of  the  Transac.  Amer.  Soc. 
Civil  Engineers,  p.  220)  on  special  forms  of  weirs;  some  of 
which  involved  a  sloping  face  on  the  up-stream  or  down-stream 
side,  or  both;  some  with  flat  tops.  Results  for  a  number  of 
these  forms  will  now  be  quoted.  Air  was  given  free  access 
under  the  sheet,  or  "  nappe,"  of  water  on  the  down-stream  face 
in  each  case. 

The  crest  of  the  weir  was  in  each  case  6.56  ft.  long  and  end- 
contractions  were  suppressed,  i.e.,  the  channel  of  approach  had 
the  same  width  as  each  weir  and  the  depth  of  water  (h^  above 
the  crest  of-  the  weir  in  some  of  these  experiments  was  in  some 
cases  as  great  as  5  ft.  The  channel  of  approach  had  the  same 
width  as  each  weir  and  was  rectangular  in  section;  and  extended 
back  some  40  ft.  from  the  weir. 

Fig.  103  gives  a  general  idea  of  the  form  of  some  of  these 
weirs  and  of  the  quantities  involved.  The  "  head  on  the  weir," 
h,  was  observed,  and  the  height  of  weir  p  was  measured  and 
recorded  in  each  case,  as  also  the  data  fixing  the  form  of  the  top 
and  two  faces  of  the  weir.  The  rate  of  flow  Q  became 
known  in  each  case  from  the  observed  head  and  known  dimen- 
sions of  a  standard  Bazin  weir  (sharp-edged  with  vertical  faces. 


§  137. 


WEIRS — SPECIAL   FORMS. 


223 


etc.)  over  which  the  water  flowed  on  its  way  to  the  experimental 
weir. 

The  formula  used  by  Mr.  Rafter  in  expressing  the  rate  of 


g^sv££2£^^ 


FIG.  103. 

flow  over  any  of  these  experimental  weirs  may  be  put  into  the 

form 

Q=mb\/r2g-(h  +  K)*}       .....     (5) 

where  6  is  the  length  of  the  weir,  m  a  coefficient  corresponding 
to  the  f  fi.  of  former  equations,  and  h  the  observed  head  on  the 


SSi'A  ^  :?:iv>'^;V::v::.i^\\:: 

A  °  n'RR  >.    «c 


i;X^^£Z£p&i^>:&&;:tt  ^^^^^^:<^^^:^^-^^:^^-^ 


FIG.   104. 


weir  (see  Fig.  103);  while  k  stands  for  c2+2g,  or  height  due  to 
the  "  velocity  of  approach/'  c,  this  velocity  being  equal  to  the 
C^area  of  cross-section  of  channel  of  approach. 

It  is  seen  that  k  depends  on  the  discharge  Q  itself;  but  it  is 


224 


HYDRAULIC    MOTORS. 


§  137. 


4:57 


generally  so  small  that  a  value  for  it  obtained  from  an  approxi- 
mate value  of  Q,  based  on  a  zero  value  for  k  in  eq.  (5),  is  suffi- 
ciently close  for  substitution  in  a  second  use  of  eq.  (5) ;  from 
which  a  second  and  closer  value  of  Q  is  secured. 

The  values  of  the  coefficient  m  will  now  be  given  as  obtained 
by  Mr.  Rafter  for  nine  weirs  of  different  shapes,  to  be  designated 
as  A,  B,  etc.     The  dimensions  and  form  of  vertical  section  of 
seven  of  these  weirs  are  shown  in  Figs.  104  and  105,  the  direc- 
tion of  flow  being  indicated 
by    the    arrow.     The    weirs 
called  D  and  E  in  the  fol- 
lowing   table    differed    from 
B  only  in  the  slope  of  the  up- 
stream   side;  which    was    4 
to  1  for  D,  and  5  to  1  for  E. 
It   will   be   noticed    that 
form  I  differs  from  H   (see 

Fig.  106)  in  being  about  one-eighth  wider  and  in  having  the  up- 
stream corner  rounded  in  a  quadrant  of  radius =0.33  ft.,  or 
4  inches;  and  it  will  also  be  noted,  from  the  table  below,  that  the 
discharge  is  thereby  increased  by  more  than  ten  per  cent.,  for 
the  lower  heads. 

This  rectangular  weir  of  broad  crest  with  rounded  up-stream 
shoulder  as  shown  in  form  I  is  also  capable  of  theoretic  treat- 
ment for  the  determination  of  discharge.  Such  treatment  has 
been  applied  by  Prof.  Unwin  in  his  article  Hydromechanics  in 
the  Encyclopaedia  Britannica  (p.  472  of  that  article).  Prof, 
Unwinds  result  gives  a  value  of  0.385  for  the  coefficient  m  of 
eq.  (5) .  With  a  slight  deduction  to  allow  for  friction,  which  has* 
been  neglected  in  Prof.  Unwin's  treatment,  this  agrees  well  with 
Mr.  Rafter's  values  for  m  for  form  I  with  the  lower  heads  (under 
2  ft.);  and  it  is  for  the  lower  heads  that  the  theory  is  more 
reasonable. 

The  following  table  gives  values  of  the  coefficient  m  (to  be 
used  in  eq.  (5))  for  six  different  values  of  the  head  h  in  feet, 
for  the  different  forms  of  weir,  A,  B,  etc.,  as  mentioned  above. 
If  a  weir  is  very  long,  as  often  occurs  with  mill-dams,  it 


§138. 


WEIRS. 


225 


&- 

0.5  ft. 

1.00 

1.5 

2 

4 

6 

A 

m  =  0.418 

0.459 

0.476 

0.470 

0.461 

0.462 

B 

m=    .401        .428 

.447 

.456 

.461 

.462 

C 

m  =    .454 

.476 

.478 

.460 

.442 

.442 

D 

in  = 

.429 

.432 

.434 

.434 

.434 

E 

m  =    .412 

.415 

.416 

.418 

.422 

.422 

F 

m=    .525 

.529 

.505 

.486 

.461 

.453 

G 

111  =    .391 

.426 

.442 

.450 

.456 

.452 

H 

m=    .324 

.333 

.343 

.354 

.400 

.433 

j. 

w  =    .369 

.375 

.378 

.384 

.422 

.442 

makes  little  difference  whether  there  is  contraction  at  the  ends 
or  not,  while  if  h  is  less  than  about  one- fifth  of  the  height  of  weir, 
p,  the  quantity  k  is  of  little  consequence;  especially  when  we 
consider  that  results  obtained  by  the  use  of  eq.  (5)  may  some- 
times be  in  error  by  two,  or  even  three,  per  cent. 

Mr.  Rafter's  paper  containing  the  account  of  the  experiments 
just  mentioned  includes  also  a  useful  resume  of  the  experiments 
made  by  Bazin  on  a  great  variety  of  weir  forms.*  See  also  pp. 
222,  etc.,  in  Turneaure  and  Russell's  " Public  Water-supplies." 

138.  Submerged  or  "  Drowned  "  Weirs. — If  the  height  of 
weir,  p,  is  less  than  the  original  depth  of  the  stream,  a  submerged 
weir  results.  But  few  experiments  have  been  made  on  this 
kind  of  weir.  According  ^£^~^= 
to  Mr.  Herschel  (Transac.  ^"^""^-r^^^j^ 
Am.  Soc.  C.  E.,  1885,  xiv,  4j^""L^7-^  - ^— -*—  /&_  ,^ 
p.  194)  for  submerged  ^^^^^^^^^^^-g 
weirs  with  sharp  crests,  ^^ 
up-stream  face  vertical,  7-^,  'P~ 
and  without  end-contrac-  ~~  — . 
tions  (as  in  Fig.  106),  the 
following  formula  may  be 
used,  based  on  the  experi-  FlG-  106> 

ments  of  Francis  and  also  those  of  Ftelev  and  Stearns: 


6 


*  Water-Supply  Paper  No.  150,  issued  by  the  U.  S.  Geol   Survey,  gives  a 
uable  resume  of  weir  coefficients,  etc.  ;  by  Robert  E.  Horton. 


valuable 


226 


HYDRAULIC    MOTORS. 


39. 


for  the  discharge  in  cub.  ft.  per  sec.;  the  length  b  of  the  weir  and 
h,  the  "head  on  the  weir/7  (see  figure,)  both  being  expressed  In 
ft.;  while  the  number  or  coefficient  n  depends  on  the  value  of 
the  ratio  hi  +  h. 

Mr.  Herschel  gives  the  following  table  for  n: 


i 

h^h 

n 

h^h 

n 

h^h 

n 

h^h 

n 

€.00 

1.000 

0.20 

0.985 

0.45 

0.912 

.70 

0.787 

.02 

1.006 

.25 

.975 

.50 

892 

,75 

.750 

.05 

1.007 

.30 

.959 

.55 

.871 

.80 

.703 

.10 

1.005 

.35 

.944 

.60 

.846 

.90 

.574 

.15 

0.996 

.40 

.929 

.65 

.819 

1.00 

.000 

Example.  —  With  the  same  stream  as  in  the  example  of 
§  136,  30  ft.  in  width,  do  the  original  depth  =3  ft.,  and  with 
0=310  cub.  ft.  per  sec.;  if  a  sharp-crested  weir  without  end- 
contractions  of  2.6  ft.  height,  =  p,  be  built  across  the  full  width 
of  the  stream,  what  increase  of  depth  will  be  occasioned  just 
above  the  weir?  That  is,  h=  ? 

Solution.  —  Since,  from  eq.  (6), 

' 


we  have,  putting  n=0.9  as  a  first  approximation,  ^=2.36  ft.; 
from  which,  since  hi+h=  (3-2.6)  -*•  2.36=  0.169,  we  find  a  value 
of  0.992  for  n,  from  the  table.  For  this  closer  value  of  n  we 
now  derive,  from  eq.  (7),  7^=2.14  ft.  as  a  second  approximation. 
Again,  hi+h  would  now  become  (3  -2.6)  -5-2.14=  0.187; 
i.e.,  from  the  table,  n  would  be  equal  to  0.988;  and  finally, 
as  sufficiently  close, 

ft-  2.  126-  0.988  =2.  15ft.; 

and  hence  the  new  depth  just  above  the  weir  will  be  p  +  h,  or 
4.75  ft. 

139.  Discontinuous,  or  Incomplete,  Weirs.     Height  of  Back- 
water. —  The  rise  of  water  in  a  stream  occasioned  by  the  building 


§  139. 


BACKWATER. 


227 


of  bridge  piers,  jetties,  moles,  breakwaters,  dikes,  or  causeways, 
may  be  approximately  computed  by  the  following  methods. 

Fig.  107  shows  the  case  of  a  jetty  projecting  part  way 
across  the  width  of  a  stream.  The  upper  part  of  the  figure 
gives  a  vertical  projection  parallel 
to  axis  of  stream,  the  lower  part  a 
horizontal  projection.  The  width  of 
the  stream,  originally  equal  to  b', 
b+e,  is  only  b,  opposite  the  end  of 
the  jetty;  which  takes  up  a  portion 
e  of  the  original  width.  This  causes 
an  increase  of  depth,  =  H,  just  above 
the  jetty  when  a  steady  flow  has 
again  set  in.  The  whole  discharge 
of  the  stream,  Q,  must  now  pass 
through  the  narrow  width  b  =  BC. 

The  fraction  of  Q  (call  it  Qi)  pass- 
ing through  the  portion  DF  of  the 
depth  may  be  treated  as  if  flowing 
through  an  overfall  notch  and  written 


FIG.  107. 


......    (8) 

while  that,  Q2,  passing  below  the  level  of  point  F  may  be  con- 
sidered as  flowing  through  a  vertical  rectangular  opening  (and 
discharging  under  water)  of  a  height  =  d0  (depth  of  tail-water; 
original  depth)  and  width  =  e,  all  filaments  having  a  common 
velocity  =\/2g-H  (p.  669,  M.  of  E.)  ;  that  is, 

Q2=  fibdoVzjH.       ......     (9) 

But  Qi+Q2=Q;  and  hence,  finally,  considering  the  two  p's  to 
be  about  equal,  we  have 

Q=l&V2gH[%H+do]  ......     (10) 

If  the  height  H  is  small  or  the  velocity  of  approach  con- 
siderable, with  k  =  c?+2g  (c  being  the  velocity  of  approach,  viz. 

c=Q+(H+do)b',    nearly)      ....    (11) 
(see  p.  674,  M.  of  E.,  eq.  (3),) 


228  HYDRAULIC    MOTORS. 

we  have,  to  take  the  place  of  eq.  (10)  above, 


§  140- 


(12) 


The  coefficient  //  may  range  from  0.70  or  0.80  to  0.95  accord- 
ing to  the  degree  of  rounding  of  the  end  of  the  dike.  In  the 
use  of  eq.  (12),  which  cannot  be  expected  to  give  more  than 
roughly  approximate  results,  it  is  best  to  solve  by  successive 
assumptions  and  trials,  to  avoid  mathematical  complications, 


FIG,   108. 


In  the  case  of  a  number  of  bridge  piers,  the  b  of  eq.  (12) 
would  represent  the  sum  of  the  widths  of  the  openings  between 
the  piers.  To  prevent  the  injurious  effects  of  eddies,  etc.,  both 
ends  of  the  horizontal  section  of  a  bridge  pier  should  be  rounded 
or  sharpened  off,  as  illustrated  in  Fig.  108.  According  to 
Weisbach  and  Gauthey,  if  the  ends  are  rounded  or  shaped  with 
a  very  obtuse  angle  /*  may  be  taken  as  0.90;  'with  an  acute 
angle,  0.95;  or  even  0.97,  if  two  circular  arcs  meeting  at  an  acute 
angle  are  used. 

140.  Amplitude  of  Backwater  caused  by  a  Weir. — The  law 
by  which  the  depth  of  the  water  which  has  been  increased  by 
the  weir  diminishes  with  the  distance  up-stream  from  the  weir 
must  now  be  investigated.  This  can  be  referred  to  the  theory 
of  steady  flow  with  variable  motion  or  "  non-uniform  motion " 
in  an  open  channel,  as  given  on  pp.  768,  etc.,  of  M.  of  E. 
Suppose  the  stream  above  the  weir  to  be  divided  into  several 
distinct  portions  by  successive  transverse  vertical  planes.  For 
each  of  these  we  may  have  a  separate  treatment,  the  surface 
of  each  being  considered  straight- by  itself.  Fig.  109  shows 


§  140. 


BACKWATER. 


229 


a  short  length  of  the  stream  above  the  weir,  the  flow  being  steady. 

Let  the  depth  at  A  be  yt  ft.,  the 

area  of  cross-section  F\  sq.  ft.,  and 

the  mean  velocity  vi  (of  stream- 

lines  passing  that  section);    also 

Wi  =  the    mean  wetted   perimeter 

of  portion  AC  of  stream.     For  sec- 

tion   C,  2/0,   FQ,  and   VQ   have   a 

similar  meaning.     The  length  AC 

call  Zi.     Let  m  and  n  be  points 

where  any  stream-line  crosses  the 

two    sections;     z\    and    -20    their 

heights  above  datum  through  0; 

and  let  water-barometer  height  be  6,  and  the  slope  of  the  bed 

BO  be  a.    Now  the  fluid  pressure  at  m  (since  flow  is  horizontal) 

/v\ 

is  atmospheric  plus  that  due  to  water  height  A.  .m,  :.  —  > 
=  pressure-head  at  m,=b+Am=b+AE—  21=  6  +2/1  -j-Zi  sin  a  —  z\\ 
and  similarly  —  =  6+2/0—20. 


For    friction-head   between   m 


and  n  we  may  use  the  form  |r  "if-  (see  eq.   (3),  p.  757,  M.  of 
E.)  with  #1  =  mean  hydraulic  radius  for  AC,  =  %[Fo  +  FI]  -5-  wi  ; 

and  Vrr?  =  %(VQ2  +  1>l2)  . 

Hence  Bernoulli's  Theorem  for  the  steady  flow  of  the  stream- 
line m  to  n  will  give  us 

f  .ft+^l'^-f.f  .  .    .    .    ,,3, 

With  above  values  of  pmj  pn,  RI,  and  vm2,  this  reduces  to 


If  eq.  (14)  be  applied  to  the  segment  of  stream  next  above  the 
weir  (see  Fig.  110),  remembering  that  the  delivery  of  the  stream, 
Q  (cub.  ft.  per  sec.,)  is=^o^o,  etc.,  so  that  v0  =  Q+F0,  Vi  =  Q+Fi, 


230 

etc.,  we  have 


Zi- 


HYDRAULIC    MOTORS. 


.Q2 
yo  -y\  - 


sma  — 


§  Ul. 


:   '.    (15) 


and  similarly  for  the  second  segment  up-stream, 

1        1  W 


fw2 


1  1Q2 


.   :   (16) 


(and  the  method  of  further  procedure  is  now  evident). 

Thus,  assuming  successive  decrements  of  depth  yo—yi, 
7/1  —y2,  etc.,  and  computing  from  these  the  areas  FI,  F2,  etc.,  we 
obtain  from  the  above  formulae  the  distances  li,l\-\-  h,  h+h  +  h, 
.  .  . ,  etc.,  from  the  weir,  of  the  sections  where  the  assumed 
depths  will  be  found. 


FIG.  110. 

141.  Numerical  Example.  Amplitude  of  Backwater.  (The 
data  are  from  Weisbach's  Mechanics,  but  the  treatment  is  more 
modern.)  It  is  required  to  determine  the  amplitude  of  back- 
water produced  by  a  weir  in  a  stream  80  ft.  wide  and  originally 
4  ft.  deep,  in  which  the  flow  was  a  uniform  motion  before  erection 
of  the  weir,  if  the  weir  causes  the  surface  (immediately  above  it) 
to  be  raised  3  ft.  higher  than  its  original  position,  and  if  the 
discharge  of  the  stream  is  Q=1400  cub.  ft.  per  sec.  The  bed 


§  141.  AMPLITUDE    OF    BACKWATER.  231 

is  of  rock,  but  fairly  smooth,  such  as  would  justify  the  use  of  a 
value  of  n=0.020  in  Kutter's  formula  for  "  uniform  motion.7' 

Before  the  erection  of  the  weir  the  slope  of  the  surface  (  =  s 
of  §  132)  was  equal  to  that  of  the  bed,  which  is  sin  a  of  our 
present  formula?.  We  shall  suppose  that  it  is  necessary  to 
compute  the  value  of  sin  a  from  a  knowledge  of  the  fact  that 
the  motion  was  "  uniform  "  before  the  erection  of  the  weir- 
Before  erection  of  weir  the  mean  velocity,  v,  was  v=  1400  -80  X4r 
=  4.37  ft.  per  sec.,  and  the  surface  of  the  water  was  parallel  to» 
the  bed,  so  that  the  relation  then  holds  (p.  757,  M.  of  E.)  (see 
also  §  132  of  this  book) 

_h        fw  v2          v2 
a'  ~  l>  ~Y'2g>  ~Z* 

Taking  Kutter's  coefficient  of  roughness,  n,  as  0.020  (see  §  132) 
we  find  that,  with  #  =  320-88  =  3.6  ft.,  the  value  of  Kutter'& 
A  (see  diagram  for  n  =  0.020  in  Appendix)  must  lie  between  91 
and  94.  With  the  value  94  for  A,  sin  a  or  s,  from  eq.  (17), 
would  be  about  .0006;  and  with  this  approximation  to  the  slope 
we  find  more  exactly,  from  the  diagram,  A  =  93:  the  use  of 
which  in  eq.  (17)  yields  a  value  of  sin  a  =  .000615,  which  will 
be  used  in  eq.  (15),  etc. 

We  have  given,  therefore,  that  yo  =  7  ft.  in  Fig.  110  and  now 
inquire,  first,  at  what  distance,  /i,  up-stream  from  the  weir,  has 
the  depth  diminished  to  y\  =  6.5  ft.  With  yQ  -  y\  =  7  —  6.5  =  0.50 
ft.,  we  have  ^0  =  80X7-  560  sq.  ft.;  ^1  =  80x6.5  =  520  sq.  ft.; 
Q  =  1400  cub.  ft.  per  sec.;  and  w\  may  be  taken  as  93  ft.  For 
this  first  portion  of  the  stream  above  the  weir  we  find  the  mean 
hydraulic  radius  R  to  be  80X6.75-93  =  5.8  ft.,  for  which  from 
the  diagram  for  n=  0.020  (in  the  Appendix)  Kutter's  coefficient 
A  is  noted  to  be  about  100;  whence  the  value  of  the  coefficient 
/,  needed  in  eq.  (15),  is,  2g  +  A2,  =0.00644. 

Substituting  now  in  eq.  (15)  we  have  (ft.  and  sec.) 

0.50  -  (.00000370  -  .00000319)30380 

~~ 


00644 
.000615  -  -(.OOOOQ370.+  .00000319)30380 

J-UoU 

0.484-0.000498  =  973  ft. 


232  HYDRAULIC    MOTORS.  §  142, 

Next  assuming  a  value  of  y  =  6  ft.,  whence  jF=480  sq.  ft.,  while 
R%,  the  mean  hydraulic  radius  of  this  second  portion  of  stream,  = 
500  -T-  92,  =  5.4  ft.  (the  mean  wetted  perimeter  being  taken  as 
w2  =  92  ft.).  For  which  R2  we  find,  from  the  diagram,  ^4  =  99, 
whence  /,  =2g+A2,  =0.00657.  Substitution  of  these,  with 
other  known  values,  in  eq.  (16)  gives  the  result  /2=  1029  ft. 

Similarly,  with  y3  =  5.5  ft.,  2/4  =  5  ft.,  and  y5  =  4.5  ft.,  we  find 
successively  Z3=1121,  Z4=1320,  and  /5=1755  ft.  That  is  to 
say,  adding  the  proper  lengths, 

The  height  of  backwater  at  the  weir  is  ................   3  .0  ft. 

"          "         "    973ft.  above  the  weir  is.  .     2.5  " 
ic       (i        ic  ic         n   900^  1  l       Ci        l  '      li     ll         *)  0  '  ' 

cc          (c  cc  (c  i(    0190   f<         (i  <(        *<      it  1     t\   {  { 

ic          c  c  c  c  (  c  (i    AAAQ   l  c         l  c          1  1        c  l      (  e  incf 

CC  CC  1C  CC  It      A1QQ     i(  <l  lt  (C         I 


This  method  must  be  looked  on  as  only  roughly  approximate. 
Theoretically  (see  next  §)  the  curve  of  backwater  is  asymptotic 
to  the  original  line  of  water  surface  (in  ordinary  cases),  so  the 
height  of  backwater  becomes  zero  only  at  an  infinite  distance 
from  the  weir. 

142.  Amplitude  of  Backwater  for  a  Shallow  Stream  of  Rect- 
angular Section.  Results  by  Calculus.  —  Consider  a  very  wide 
stream  of  rectangular  section,  in  which  the  depth  was  =  do 
(same  at  all  sections)  and  motion  uniform,  velocity  =  v,  = 
Q  +  bdo,  before  the  erection  of  a  weir;  width  =  6  at  all  sections, 
before  and  after  erection  of  weir.  Fig.  Ill  shows  a  profile 
LMY  of  water  surface  after  erection  of  weir,  also  original  posi- 
tion TR  of  surface.  Take  axes  X  and  Y,  as  shown.  Then,  at 
any  point  M  of  profile,  y  =  MP  is  depth  of  water  and  y'  =  MR 
is  the  increase  of  depth  (or  "height  of  backwater  "),  due  to  the 
weir,  at  any  distance  x  =  PO  from  the  weir.  The  slope  of  the 
bed  will  be  denoted  by  sin  a,  and  the  quantity  v2+  2g,  or  height 
due  to  the  original  mean  velocity,  by  k. 

If  the  foregoing  treatment  of  successive  finite  reaches  along 
the  stream  above  the  weir  be  applied  to  vertical  slices  or 
laminae  of  horizontal  thickness  dx,  the  areas  of  the  two  faces  of 


§  142- 


AMPLITUDE    OF   BACKWATER.      EXAMPLE. 


233 


each  such  lamina  being  by  and  b(y—dy),  a  differential  equation 
may  be  formed  and  an  integration  effected,  involving  one  or 
two  approximations  (detail  will  be  found  in  the  works  of  Weis- 
bach,  Bresse,  and  Grashof,  etc.),  resulting  in  the  following 


WEIR 


HORIZONTAL 


FIG.  111. 


equation  to  the  curve  of  backwater  in  this  case,  L  .  .  .  Y,  or 
profile  of  water  surface  after  the  erection  of  the  weir,  viz. : 


(sma)x=y0  —  y  +  (do-2k)((/)-<l)o);  .     .     .     (18) 

x  and  y  being  the  two  coordinates  of  any  point  M  on  the  curve, 
and  7/0  the  depth  of  the  water  immediately  above  the  weir; 
while  $  is  a  variable  (and  abstract  number)  and  a  function  of 
the  ratio  y  +  do.  A  series  of  values  of  <£,  sufficient  for  practical 
purposes,  is  here  given: 

TABLE  OF  VALUES  OF  THE  FUNCTION  <j>. 


For  f-  =     1.0      1.1      1.2 

do 


1.3       1.4       1.5       1.6       1.7 
infinite  .680    .480    .376    .304    .255    .218    .189 


For  ~  -     1.8      1.9      2.0      2.2      2.4      2.6      2.8      3.0 

dQ 

<£=  .166    .147    .132    .107    .089    .076    .065    .056 

By  <£o  in  (18)  is  meant  the  value  of  <£  when  y=yo. 

Hence,  for  a  shallow  stream  of  rectangular  section,  given 
the  data  of  the  weir,  height  and  backwater  at  the  weir,  original 


234  HYDRAULIC    MOTORS.  § 

depth  of  stream,  do,  the  constant  width,  6,  etc.,  we  can  find  the 
"amplitude"  x,  up-stream  from  the  weir,  of  the  point  where 
any  assumed  depth  y  [or  height  of  backwater  y  —  do]  will  be 
found.  Of  course  y  must  lie  between  yo  and  do  in  value. 

For  y  =d0,  evidently  x=  <*>  (see  table). 

To  justify  the  use  of  this  method  the  depth  must  not  be 
greater  than  about  one-fifteenth  of  the  width  6. 

Example. — Let  us  apply  the  foregoing  table  for  <£  and  eq. 
(18)  to  the  data  of  the  example  just  treated  in  §  141. 

As  already  found  in  that  paragraph,  sin  a  =0.000615;  while 
&=  (4.37)2 -H  2^  =  0.34  ft.  Since  y0  •*-*>  =  7*  4 -1.75,  we  have, 
from  the  table,  ^o^  0.177.  Let  us  inquire  the  value  for  x  in 
order  that  y  may  be  equal  to  6.5  ft.;  that  is,  y~do  =  6.5-^4= 
1.625;  for  which  from  table  we  find  <£  =  0.211. 

Substitution  of  these  values  gives 

(0.000615)o;=  7  -6.5  +  (4-0.68)  (0.211  -0.177); 

whence  £  =  997  ft. 

Again,  the  distance  x  from  the  weir  at  which  the  new  depth 
will  be  y  =  5.5  ft.,  ^  being  now  found  to  be  0.322,  is  determined 
by  solving  the  equation 

(0.000615)z=  7-5.5  +  (4 -0.68)  (0.322 -0.177); 

i.e.,  z  =  3225  ft. 

It  is  seen  that  these  results  do  not  differ  greatly  from  those 
found  by  the  more  cumbersome  method  of  §  140. 

•  143.  Other  Equations  for  the  Backwater  Curve. — According 
to  Poiree,  the  backwater  curve  may  be  considered  to  be  a 
parabola  with  its  axis  vertical  and  having  its  vertex  at  Y  (see 
Fig.  HI);  its  actual  magnitude  in  any  case  being  determined 
by  making  it  tangent  to  the  original  surface  TR  at  a  point 
whose  abscissa  is  x=2H+sma,  with  H  denoting  y—do  (as 
marked  in  figure)  or  increase  of  depth  at  the  weir.  That  is, 
the  equation  to  the  parabola  of  backwater  would  be,  on  this 
basis, 

(sin2  a^x2 

— .     .     .     (19) 


§144. 


VARIABLE    MOTION.      STEADY   FLOW. 


235 


St.  Guilhem's  equation  to  the  backwater  curve  is  much  more 
complicated,  but  was  devised  to  correspond  as  nearly  as  possible 
to  actual  measurements  made  of  a  backwater  curve  on  the 
river  Weser,  in  Germany.  (See  p.  772,  M.  of  E. ;  and  Bennett's 
translation  of  D'Aubuisson's  Hydraulics,  pp.  179  and  180). 
The  formula  is  (for  the  English  foot) 

H* 


(a; -sin  a)6 

7.382 


(20) 


(Eqs.  (19)  and  (20)  are  for  use  with  shallow  streams.) 

Other  references  in  this  connection  are :  Engineering  Record, 
July  1892,  p.  91 ;  Report  of  Chief  of  Engineers  of  U.  S.  Army 
for  1887,  p.  1305;  DeBauve's  Manuel  de  Plngenieur;  Annales 
des  Fonts  et  Chaussees  for  1837;  Transac.  Am.  Soc.  Civ.  Engs., 
vol.  ii,  p.  255. 

144.  Variable  Motion,  but  with  Diminishing  Depth.  Steady 
Flow. — In  the  foregoing  numerical  illustrations  of  variable 
motion  the  cases  have  been  those  of  depth  increasing  going  down- 
stream, since  the  backwater  due  to  a  weir  was  under  treatment. 
We  now  take  an  example  in  which  the  depth  diminishes  down- 
stream, see  Fig.  112.  Let  the  open  channel  have  bottom  and 


FIG.  112. 


sides  of  rough  brickwork  (or  n  =  0.015  as  Kutter's  coefficient 
of  roughness),  with  width  6  =  10  ft.  and  length  600  ft.  and  a 
horizontal  bed.  It  connects  a  large  pond  A  with  another  pond 
(or  wheel-pit)  E.  When  the  positions  of  the  surfaces  of  the 


236  HYDRAULIC   MOTORS.  §   144. 

water  in  these  two  ponds  are  such  that  depth  in  the  channel  is 
2.50  ft.  at  C  (an  allowance  having  been  made  for  the  small 
drop  EC  of  the  surface)  and  is  2.20  ft.  at  D,  it  is  required  to 
compute  what  the  rate  of  discharge  (Q)  must  be,  under  these 
circumstances.  The  cross-section  of  channel  is  rectangular. 

Since  an  approximate  value  for  the  mean  hydraulic  radius 
is  10X2.4-^14.8  =  1.6  ft.  (w\,  the  mean  wetted  perimeter, 
being  10+2x2.4,  =  14.8  ft.);  the  average  slope  of  the  surface 
also  being  0.3-^600  or  0.5  ft.  per  thousand;  we  find  in  the 
diagram  for  n  —  0.015  (in  Appendix)  the  value  A  =  107.  From 
this  we  have  for  the  coefficient  of  friction  /,  =2g  +  A2,  =0.00562. 

Taking  the  whole  length  of  600  ft.  as  a  single  reach  we  may 
now  apply  eq.  (15)  of  §  140  with  values  above  obtained  for  /  and 
Wi  and  also  7/0  =  2.2  and  7/1  =2.5  ft.;  sin  a  =  zero;  Fo  =  22  and 
Fi  =  25  sq.  ft.;  with  ^=600  ft.  Careful  attention  being  paid 
to  the  signs,  we  finally  derive  Q  =  66.4  cub.  ft.  per  second. 

This  value  of  Q  implies  a  mean  velocity  at  section  C  of 
2.65  ft.  per  second.  For  the  water  to  acquire  this  velocity 
at  C  the  surface  must  fall  a  vertical  distance  BC=  (2.65) 2  +  2g 
=  0.109  ft.;  so  that  the  whole  difference  of  level  between  the 
surfaces  of  still  water  in  the  two  ponds  must  be  0.109+0.30, 
=  0.409  ft.,  if  the  above  rate  of  discharge  is  to  take  place. 

It  will  be  of  interest  to  note  that  if  the  bottom  (starting  at 
the  same  point  F)  were  given  a  downward  slope  parallel  to  that 
which  it  is  desired  that  the  surface  of  water  shall  have  (that  is, 
drop  0.30  ft.  in  the  600  ft.  of  length),  we  have  a  case  of  "  uniform 
motion"  to  which  Kutter's  formula  may  be  applied  (see  §  132; 
A  being  taken  from  the  proper  diagram).  The  result  for  the 
discharge  is  then  found  to  be  Q  =  75  cub.  ft.  per  sec.;  which  is 
greater  (as  of  course  it  should  be)  than  in  the  first  case.  The 
mean  velocity,  then,  in  all  sections  would  be  3  ft.  per  sec.,  and 
the  drop  BC  would  be  0.15  ft.;  necessitating  a  difference  of 
level  from  surface  A  to  surface  E  of  0.15  +  0.30=0.45  ft. 

It  must  be  remembered  that  in  all  problems  of  this  class 
there  is  considerable  uncertainty  as  to  the  influence  of  the 
roughness  of  the  bed  which  cannot  be  brought  into  play  with 
any  precision. 


§  145.  VARIABLE    MOTION.      STEADY   FLOW.  237 

145.  Open  Channel  of  Horizontal  Bed  and  Shallow  Depth. 
Depth  Diminishing  Down-stream.  —  In  case  the  depth  diminishes 
down-stream  in  steady  flow  in  an  open  channel  of  rectangular 
section  with  horizontal  bed  and  shallow  depth  (the  depth  not 
being  greater  than  (say)  one-fifteenth  of  the  width)  ,  an  applica- 
tion of  the  calculus  to  the  successive  vertical  laminae  (of  horizon- 
tal thickness  dx)  leads  finally  to  the  relation  (adapted  from 
p.  454  of  Ruehlmann's  Hydromechanik) 


dtf',       .     .     .     (21) 

in  which  b  is  the  (constant)  width  of  the  channel  (rectangular 
section),  d  the  depth  at  any  point,  and  d\  the  depth  at  any 
distance  I  down-stream  from  the  first;  while  Q  is  the  volume 
carried  per  second.  The  coefficient  of  fluid  friction,  /,  may  be 
obtained,  as  previously,  from  the  relation  f  =  2g+  A2  (see  §  132, 
eq.  (4)),  where  A  is  Kutter's  coefficient,  to  be  found  from  the 
diagrams  in  the  Appendix. 

The  quantities  d  and  I  may  be  looked  upon  as  ordinate  and 
abscissa  (see  Fig.  112),  with  K  as  an  origin,  of  the  curve  formed 
by  the  upper  longitudinal  profile  of  the  water  surface.  In 
applying  this  relation  the  restriction  as  to  the  depth  being  small 
should  be  carefully  borne  in  mind;  since  otherwise  results 
might  be  obtained  which  would  be  very  wide  of  the  truth. 
(The  example  worked  out  in  the  preceding  problem  could  not 
be  treated  by  eq.  (21),  as  the  relative  depth  is  much  too  great.) 

Example.  —  Let  the  width  of  the  channel  (with  horizontal 
bed;  see  Fig.  112,  which  will  serve  our  present  purpose,  although 
the  numerals  there  printed  do  not  now  apply)  be  100  ft.,  and 
the  depth  at  section  C  be  d=3  ft.  If  it  is  noted  that  the  depth 
di  at  section  D,  1000  ft.  down-stream  from  C,  is  2  ft.,  at  what 
rate  must  the  water  be  flowing  (volume  per  sec.,  Q=?)  Let 
the  degree  of  roughness  of  the  bed  be  such  as  to  correspond 
to  Kutter's  n=  0.020. 

The  mean  slope  (with  reference  to  finding  /)  may  be  taken 
as  1  ft.  per  thousand  and  the  mean  hydraulic  radius  R  as  2.25 
ft.  With  these  values  in  the  diagram  for  n  =  0.020  (Appendix) 
we  find  .A  =85;  and  therefore  /,  =2g+  A2,  =0.00891. 


238 


HYDRAULIC    MOTORS. 


§  146. 


Substitution  in  eq.  (21)  gives 

2Q2  [8.91  +  2(3  -  2)]  =32.2(100)2(81  - 16)  ; 

from  which,  finally,  Q=980  cub.  ft.  per  sec. 

(Note. — If  in  this  case  the  bed  sloped  parallel  to  the  surface 
of  the  water,  the  depth  being  3  ft.  at  all  sections,  we  should 
have  a  case  of  "uniform  motion/'  to  which  Kutter's  Formula 
would  be  applicable ;  and  the  result  in  that  case  would  be  found 
to  be  Q  =  1290  cub.  ft.  per  second.) 

146.  Standing  Waves. — A  "  standing  wave,"  or  "  hydraulic 
jump,"  may  be  formed  by  the  introduction  of  an  obstruction  in 


i._ 

HORIZONTAL 


FIG.  113. 


a  stream  whose  velocity  c  is  so  great   (relatively)   that  k,  or 
c*+2g,  is  greater  than   one   half    the   depth,  dQ.      Fig.    113. 


Fro.  114. 


Other  considerations  also  enter.  For  the  mathematical  treat- 
ment involved,  see  Ency.  Britann.,  article  Hydromechanics, 
p.  501;  Weisbach's  Hydraulics  and  Hydraulic  Motors,  §  154; 


§   146.  OPEN    CHANNEL.      HORIZONTAL    BED.  239 

Ruehlmann's  Hydromechamk,   p.   475;    and  also  Merriman's 
Hydraulics,  p.  342.     (And  Eng.  News,  July  1895,  p.  28.) 

A  change  of  slope  in  the  bed  (Fig.  114)  may  also  occasion  a 
standing  wave. 


CHAPTER  IX. 
PRESSURE-ENGINES,  ACCUMULATORS,  AND  HYDRAULIC  RAMS. 

147.  Pressure-engines. — In  Fig.  115  we  have  a  previous 
figure  reproduced  (see  §  6)  and  now  take  up  more  fully  the 
subject  of  water-pressure  machinery.  In  a  water-pressure 
engine  we  have  in  general  a  piston  capable  of  a  reciprocating 
(to  and  fro)  motion  in  a  cylinder,  the  edges  of  the  piston  fitting 
accurately,  allowing  no  leakage  of  water.  At  the  two  ends  of 


FIG.  115. 

the  cylinder  are  ports  or  passageways,  opened  and  closed  at 
the  proper  time  by  sliding  pieces  called  valves  (or  if  cylindrical 
in  shape,  like  stoppers,  then  piston-valves).  In  this  way 
either  side  of  the  end  of  the  cylinder  may  be  put  into  com- 
munication with  the  supply-pipe  r  or  with  the  exhaust-pipe 
leading  to  the  tail-water  T  (or  directly  to  the  outer  air).  In 
case  the  tail-water  is  below  the  level  of  the  cylinder  the  exhaust- 
pipe  is  called  a  suction-tube  or  draft-tube.  In  that  case  the 
height  of  the  cylinder  above  the  tail-water  is  limited  to  about 
20  or  25  ft. 

240 


§  147.  WATER-PRESSURE    ENGINES.  241 

A  motor  is  said  to  be  "single-acting"  when  the  " exhaust 
side/'  ra',  is  always  open  to  the  atmosphere,  while  the  other 
side,  ra,  communicates  alternately  with  the  outer  air  or  with 
the  source  of  supply.  In  a  single-acting  pressure-motor  no 
working  force  acts  in  the  return  stroke,  i.e.,  no  useful  work 
is  done,  the  motion  being  brought1  about  by  the  inertia  of  a 
fly-wheel  or  by  the  action  of  some  other  working  piston  if 
there  are  more  than  one  piston  and  cylinder  provided. 

A  "double-acting"  motor  is  one  in  which  during  the  return 
stroke  the  two  ends  of  the  cylinder  change  places  as  regards 
communicating  with  the  supply  E  or  with  the  tail- water  (or 
exhaust)  T.  In  such  a  case  about  the  same  amount  of  useful 
work  is  done  in  the  return  as  in  the  forward  stroke;  though 
account  must  be  taken  of  the  fact  that  there  is  a  difference 
between  the  areas  of  the  two  faces  of  the  piston,  since  on  one 
side  the  sectional  area  of  the  rod  must  be  subtracted  from 
that  of  the  full  circle  of  the  piston  face. 

In  the  simple  case  in  Fig.  115  no  valves  are  shown  and  the 
supply-pipe  is  very  short,  so  that  if  a  proper  resistance  Rf  is 
provided  in  the  piston-rod  the  piston  will  move  very  slowly 
and  the  pressure  in  ra  remain  nearly  equal  to  that  in  the  still 
water  at  E  (hydrostatic  value) ;  and  similarly  the  pressure  at 
point  ra'  will  be  but  slightly  in  excess  of  that  due  to  its  depth 
below  the  surface  of  T.  When  the  piston  reaches  the  right- 
hand  end  of  its  stroke  (if  the  engine  is  "  double-acting  ")  the 
valves  are  automatically  moved  in  such  a  way  as  to  admit 
the  "  pressure- water  "  from  E  to  the  right-hand  face  of  the 
piston,  while  shutting  off  that  end  of  the  cylinder  from  T7; 
and  simultaneously  the  port  leading  to  the  left-hand  side  of 
the  piston  is  thrown  into  communication  with  T  and  that 
with  E  is  shut  off.  It  is  also  arranged  that  the  resistance  Rr 
shall  reverse  its  direction  during  this  return  stroke. 

Usually  the  cylinder  is  not  very  near  to  the  reservoir  W, 
and  a  supply-pipe  is  necessary  to  conduct  the  water  to  the 
motor.  During  the  motion  of  the  piston  the  water  in  this 
pipe  has  a  certain  amount  of  headway,  i.e.  velocity,  with  corre- 
sponding energy  of  motion,  so  that  to  prevent  "water-hammer  " 


16 


242  HYDRAULIC   MOTORS.  §  148. 

when  the  piston  stops  at  the  end  of  the  stroke  an  air-vessel  is 
provided  at  the  down-stream  end  of  the  pipe  and  communicat- 
ing with  it.  When  the  piston  stops,  some  of  the  water  flows 
into  the  air-vessel,  compressing  the  air  somewhat  more  than 
before,  and  the  velocity  of  the  water  in  the  pipe  is  thus  gradually 
destroyed,  or  perhaps  only  partially  checked,  before  the  reverse 
motion  of  the  piston  permits  new  acquirement  of  velocity. 
Frequently  water-pressure  engines  are  built  in  pairs,  one  engine 
working  the  valves  of  the  other,  in  which  case,  if  large  air- 
vessels  are  provided,  the  motion  of  the  water  in  the  supply-pipe 
is  almost  a  "  steady  flow/'  instead  of  intermittent  in  velocity. 
Since  water  is  not  highly  compressible  like  steam,  much  larger 
ports  must  be  provided  than  for  steam-engines,  to  avoid  losses 
of  head. 

If  the  piston-rod  is  connected  to  the  crank  of  a  shaft  and 
fly-wheel,  the  motion  of  the  piston  is  nearly  harmmic,  the 
maximum  velocity  occurring  at  mid-stroke.  With  a  long 
supply-pipe  without  air-vessel  the  velocity  of  the  water  in  the 
pipe  would  be  of  corresponding  character,  and  the  pressure  felt 
by  the  piston,  if  there  were  no  fluid  friction  in  the  pipe,  would 
be  least  near  the  beginning  of  the  stroke,  while  the  water  is 
gaining  velocity,  attain  its  average  value  at  mid-stroke,  and 
reach  its  maximum  toward  the  end,  when  the  previously  acquired 
velocity  of  the  water  is  checked  and  finally  reduced  to  zero; 
so  that  the  final  pressure  against  the  piston  is  greater  than  the 
hydrostatic;  but  on  account  of  fluid  friction,  which  increases 
nearly  as  the  square  of  the  velocity,  and  also  on  account  of 
the  use  of  an  air-vessel,  the  fluctuation  of  pressure  in  actual 
practice  is  less;  the  least  pressure  being  at  about  mid-stroke; 
the  final  pressure  is,  however,  still  the  greatest. 

148.  Direct-acting  Pressure-motor  and  Pump. — Referring  to 
Fig.  115  (in  which  A  and  B  are  open  piezometers),  let  us  suppose 
that  through  the  use  of  air-vessels  and  a  long  stroke  for  the 
piston,  with  small  velocity  /,  a  practically  steady  flow  is 
realized  in  the  supply-  and  discharge-pipes,  so  that  Bernoulli's 
Theorem  may  be  applied;  with  a  constant  working-force  on 
the  piston.  This  working-force,  if  pm  =  (b+y)r  and  pm'  =  (&  +  2/0  r 


§  148.  WATER-PRESSURE    ENGINES.  243 

(b  is  the  height  of  the  water-barometer)  are  the  unit-pressures 
on  the  two  faces  of  the  piston,  will  be  F(pm—pm>)  Ibs.,  and 
for  the  equilibrium  of  the  piston  in  its  uniform  motion  we  have 

F(pm-pm,)=R'+RQ',  (Ibs.),      .    .    .    .     (1) 

in  which  F  is  the  area  of  piston  (considered  same  on  both 
sides),  R'  the  thrust  (or  pull)  in  the  piston-rod,  and  R0  the 
total  friction  of  edge  of  piston  and  sides  of  rod  on  the  walls 
and  stuffing-box  of  the  motor  cylinder. 

Let  there  be  a  long  supply-pipe  (E  to  r)  of  length  I  and 
diameter  d,  in  which  the  loss  of  head  is  hp'}  and  a  discharge- 
pipe  leading  from  ra'  to  reservoir  T,  for  which  we  have  similarly 
/',  d'j  and  hp';  and  let  HE  denote  the  entrance-loss  of  head, 
at  E,  of  supply-pipe,  and  hr  the  loss  of  head  due  to  passage 
through  port  at  r;  and  h/  that  due  to  port  leading  to  discharge- 
pipe.  Also  let  Hm  and  Tm'  denote  the  vertical  heights  between 
points  involved.  Then  Bernoulli's  Theorem  applied  between 
points  H  and  ra  leads  to  the  relation 

22-b+Hm-hx-hr-h,; (2) 

and  similarly,  between  ra'  and  T, 

T)__,  

(3) 


(The  velocity-heads  at  ra  and  m'  are  ignored,  as  very  small.) 
Now  the  work  done  per  sec.  (power)  by  the  force  F(pm—pm') 
on  the  piston-rod  is  [see  (1)]  F(pm—pm')v';  i.e., 

R'v'+RQvf,  =Fv'(pm-pm,),  =Q(pm-pm>};    .    .     (4) 

in  which  Q  is  the  rate  of  discharge  (cub.  ft.  per  sec.)  through 
the  motor.  Substituting  from  (2)  and  (3),  however,  noting 
that  Hm—Tm'=ihe  whole  head  h  of  the  "mill-site,"  we  have 
(ft.-lbs.  per  sec.) 

R'v'+R0v'=Qrfh-(hE  +  hr  +  hF  +  hr'+hF')].     .    .     (5) 

(Note. — The    quantity    in    the    bracket=/^i,    the    vertical 
distance  between  piezometer  summits.) 


244  HYDRAULIC    MOTORS.  §  148. 

In  passing,  we  may  note  that  if  R'  and  R0  were  zero,  the 
bracket  would  be  zero;  which  gives  h  =  hE  +  hr  +  hF  +  hr'  +  hF', 
or  h  =  I  (friction-heads);  and  the  speed  of  steady  flow  then 
attained  (with  very  long  pipes)  would  not  necessarily  be  exces- 
sive, since  losses  of  head  vary  with  the  square  of  the  velocity,, 
nearly.  This  illustrates  the  fact  that  in  such  a  case  the  motor 
and  pipes  contain  a  hydraulic  governing  action  within  them- 
selves, preventing  large  changes  of  speed  when  a  change  takes 
place  in  the  value  of  the  resistance  Rf ' .  For  instance,  if  through 
a  diminution  in  R'  (Ibs.)  we  note  that  the  velocity  of  the  piston 
has,  after  adjustment  to  steady  flow,  been  increased  by  ten 
per  cent.,  the  vertical  drop  from  H  to  summit  A  would  be  found 
to  have  increased  by  some  twenty  per  cent.,  thereby  diminishing 
the  pressure  on  the  left  face  of  the  piston  and  providing  the 
smaller  working  force  called  for  by  the  diminution  in  /£';  and 
there  is  no  further  increase  in  speed. 

As  to  the  special  speed  which  would  cause  the  power 
L=F(pm—pm>)v'  to  be  a  maximum;  in  such  a  case  (very  long 
pipe)  it  may  be  proved  by  the  calculus  that  it  is  the  speed 
corresponding  to  the  relation  that  one-third  of  the  head  h 
shall  equal  the  aggregate  friction-head  (nearly);  but  the  con- 
sumption of  water  which  such  a  result  would  carry  with  it 
might  be  far  in  excess  of  the  capacity  of  the  "mill-site." 

As  regards  the  useful  purpose  for  which  the  motor  is  used, 
let  us  now  suppose  that  the  other  end  of  the  piston-rod  is- 
attached  to,  and  operates,  the  piston  of  a  force-pump  (not 
shown  in  figure),  this  piston  moving  horizontally  in  a  cylinder 
with  ports  and  valves  enabling  each  of  its  extremities  to  com- 
municate alternately  with  an  inlet-pipe,  of  length  l"r  and 
diameter  dfn ',  conducting  water  from  a  well  or  other  source  of 
supply,  and  with  a  delivery-pipe  of  length  /iv  and  diameter  c?v. 
We  shall  assume,  also,  that  by  the  use  of  air-vessels,  etc.,  a 
practically  steady  flow  takes  place  through  these  two  pipes- 
and  the  pump,  by  whose  operation,  maintained  by  the  motor, 
water  is  pumped  in  steady  flow,  at  the  rate  of  Q'"  cub.  ft.  per 
sec.,  through  a  height  hm  from  source  of  supply  to  surface  of  a 
receiving-reservoir.  Let  the  loss  of  head  at  entrance  of  inlet- 


§  149.  WATER-PRESSURE    ENGINES.  245 

pipe,  and  those  in  the  two  pipes,  and  ports  of  pump,  be  denoted 
by  hEm,  V",  V",  h™,  and  hFiv,  respectively.  At  least  one  of 
the  pipes  is  very  long,  so  that  the  friction-head  in  it  is  much 
larger  than  that  from  all  other  sources.  As  before,  velocity- 
heads  will  be  neglected;  and  by  the  employment  of  Bernoulli's 
Theorem,  as  previously  for  the  motor  itself,  we  may  easily 
derive  an  expression  for  the  power  exerted  in  operating  the 
pump  (for  which  R'  now  acts  as  a  working  force),  viz.: 

fiV  =#0"V  +Q"'rW"  +  W"-+h/"4-biP'+W+hMi  (6) 

in  which  RQ"  is  the  total  friction  on  the  piston  edges  and  on 
sides  of  piston-rod  (in  pump-cylinder  and  its  stuffing-box). 
(N.B.  In  case  the  delivery-pipe  of  the  pump  terminates  in 
a,  nozzle  to  form  a  fire-jet  in  the  open  air,  of  velocity  v  ft.  per 
sec.,  the  nozzle  being  at  an  elevation  hv  ft.  above  the  well; 
then,  in  place  of  the  h'"  of  eq.  (6),  we  should  substitute 

Av-f(l+— j— ;  since  in  such  a  case  the  velocity-head  in  the 

jet  would  be  of  great  importance.) 

149.  Numerical  Example  of  Foregoing  Pump  and  Motor. — 

It  is  required  to  pump  Q'"  =0.42  cub.  ft.  per  sec.  through  a 
height  of  hm  =100  ft.,  both  the  delivery-pipe  and  inlet-pipe  (or, 
it  may  be,  suction-pipe)  being  4  in.  in  diameter,  and  their  lengths 
Ziv  =  1200  ft.  and  /'"=40  ft.  (this  means,  from  the  friction- 
head  diagrams  in  Appendix,  friction- head  at  rate  of  28  ft.  per 
thousand  feet  length). 

The  available  head  for  the  pressure-engine  is  h =40  ft.;  the 
length  of  its  12-inch  supply-pipe  2000  ft.,  and  that  of  its  dis- 
charge-pipe 25  ft.  (also  12-inch).  Determine  the  necessary 
rate  of  discharge  Q,  of  water  used  by  the  motor,  for  the  opera- 
tion of  the  pump.  Consider  all  pipes  to  be  "clean  cast-iron 
pipe." 

From  the  friction-head  diagrams  we  find  V'  =  ruihF  °f 
28,  =1.12  ft.  and  Vv  =  ff#*  of  28,  =33.6  ft. 

We  have  also  Q^=26.25  Ibs.  per  sec. 

Assuming  that  the  other  losses  of  head  in  eq.  (6)  are  about 
4  ft.,  and  that  RJ"  is  TV  of  #',  we  have,  from  (6), 

0.9#V=26.25[100  +  1.12+33.6  +  4];     .     .     .     (7) 


FIG.  lisa.     Worthington  Water-motor  Pump. 
(See  foot-note  on  page  249. ) 


246 


§  150.  WATER-PRESSURE  ENGINES.  247 

that  is,  #V  =  (26.25X138.7)  -0.9=4046  ft.-lbs.  per  sec.  This 
power,  R'v'j  must  be  furnished  by  the  motor  to  operate  the 
pump,  and  with  R'v'  known  we  must  now  find  Q  from  eq.  (5), 
But  here  we  are  met  by  the  difficulty  that  the  friction-heads^ 
depend  on  the  velocity  of  the  steady  flow  in  the  pipes,  that 
is  upon  Q  itself  (diameters  being  already  fixed) .  It  is  therefore- 
necessary  to  solve  by  trial.  Since  with  no  friction  of  any  kind! 
we  should  have  Qrh=Q'"rh'",  Q  will  (very  roughly)  be  three 
times  Q'" ';  let  us  say  six  times,  to  allow  for  friction;  i.e.,  assume 
for  the  first  trial  Q  =6X0.42  =2.52  cub.  ft.  per  sec. 

From  the  friction-head  diagram  for  12-inch  clean  cast- 
iron  pipe  we  find,  for  Q=2.52,  /^=finnr  of  3-5,  =7  ft.,  while 
V^-nnhr  of  3-5>  =0.087  ft.  (neglect);  and  shall  assume  the 
remaining  losses  of  head  in  eq.  (5)  as  aggregating  2.0  ft.  Putting 
these  values  into  eq.  (5)  and  taking  Ro=^  of  R' ',  we  have 

W=Qx62.5[40-(7  +  2)], (8) 

and  hence  Q  =  (1.1X4046)  ^ (62.5X31)  =2.3  cub.  ft.  per  sec.; 
which  is  smaller  than  the  assumed  2.50.  A  second  trial  with 
()  =2. 10  is  practically  confirmed  by  eq.  (5),  and  this  result  will 
be  adopted. 

We  next  choose  a  small  value  for  i/,  say  1  ft.  per  second, 
and  assume  that  there  is  no  leakage  around  the  edge  of  either 

i2 

piston  (no  "slip,"  as  it  is  called) ;  whence,  writing  -j-  =2.1  -M/, 

we  find  the  proper   diameter  of  the  motor  cylinder  to  be 

nd'"2 
d  =  1.64  ft.,  or  close  to  20  inches;   and  similarly  with  — r—  = 

0.42H-1/,  that  the  diameter  of  the  pump  cylinder  should  be 
0.732  ft.,  or  8.78  inches. 

These  results  can  by  no  means  be  looked  on  as  exact  on 
account  of  the  uncertainty  as  to  the  loss  of  head  in  the  ports 
and  the  frictions  R0  and  RQ"  in  the  cylinders  and  stuffing- 
boxes;  but  will  help  to  give  a  clear  idea  of  the  quantities  con- 
cerned in  the  problem  and  of  the  method  of  solution. 

150.  The  Worthington  Water-motor  Pump. — The  Worthing- 
ton  Pumping  Engine  Co.  of  New  York  City,  London,  etc., 


FIG.  116. 


FIG.  117. 


248 


§  151.  WATER- PRESSURE  ENGINES.  249 

manufacture  a  "  water-motor  pump  "  of  the  type  described  and 
illustrated  in  §§  148  and  149;  i.e.,  a  pair  of  engines,  each  direct- 
connected  to  a  pump.  To  quote  from  the  award  received  at 
the  World's  Columbian  Exposition  at  Chicago  in  1893: 

"Two  pressure-cylinders,  worked  with  water-pressure  at 
one  end,  drive  two  pumps  at  the  other.  The  valve  motion  is 
a  tappet  motion,  one  piston-rod  giving  motion  to  the  valve 
on  the  other  cylinder.  It  may  be  regarded  as  an  hydraulic 
relay,  or  pressure  intensifier.  It  is  a  very  useful  appliance 
and  can  be  employed  to  pump  water  at  a  long  distance,  instead 
of  having  an  isolated  steam-plant,  which  would  require  more 
attention,  and,  in  many  cases,  an  extra  boiler  and  attendant." 

In  Fig.  115a  is  shown  a  longitudinal  section  of  one  of  these 
Worthington  water-motor  pumps.* 

151.  The  Brotherhood  Pressure-engine. — A  cross-section  of 
this  motor  at  right  angles  to  the  shaft  is  shown  in  Fig.  116. 
It  includes  three  working  cylinders,  A,  B,  and  (7,  set  at  120° 
apart,  forming,  with  their  pistons,  three  distinct  motors,  each 
of  which  is  tl  single-acting/'  one  side  of  each  piston  being  always 
open  to  the  atmosphere  at  m.  In  cylinder  A,  for  instance, 
when  the  piston  n  moves  out  (down,  in  the  figure)  pressure- 
water  is  entering  the  space  a  through  the  port  e  and  the  piston, 
through  its  connecting-rod  c,  is  exerting  a  thrust  against  the 
crank-pin  r,  which  revolves  continuously  in  the  circle  rmc, 
and  causes  rotation  of  the  main  shaft.  On  the  return  stroke, 
the  port  e,  by  movement  of  the  proper  valve,  is  opened  to  the 
atmosphere  and  the  pressure  is  equalized  on  the  two  sides 
of  the  piston  (except  for  friction  of  the  piston  in  the  cylinder) 
and  the  water  expelled.  The  piston  has  leather  packing  around 
the  edge.  It  is  evident  that  with  this  arrangement  of  three 
pistons  and  cylinders  the  engine  is  always  ready  to  start  and 
cannot  be  "stalled"  on  a  "dead-center";  since  at  least  one 
of  the  pistons  is  always  in  a  position  to  exert  a  thrust  against 
the  crank-pin.  With  a  single  supply-pipe  feeding  all  three 
cylinders  the  flow  in  the  pipe  is  fairly  "steady"  although  the 

*  Fig.  115a  shows  a  special  design  where  the  piston  of  the  "  power  end," 
at  the  left,  is  of  smaller  area  than  that  of  the  pump,  at  the  right;  for  use  where 
impure  water  from  a  high  elevation  is  used  to  pump  purer  water  against  a 
small  head. 


250  HYDRAULIC  MOTORS.  §   153. 

v. 

motion  of  each  piston  is  variable.  This  engine  is  made  in 
England  and  runs,  when  necessary,  at  high  speed;  and  with 
very  good  efficiency.  It  operates  its  own  valves. 

152.  The  Schmidt  Oscillating  Engine. — Fig.  117  (taken  from 
Knoke's  Kraftmaschinen  des  Kleingewerbes,  1887)  shows  a  cross- 
section  of  this  engine  at  right  angles  to  the  main  shaft.    There 
is  no  connecting-rod;     the  piston-rod  being  attached  directly 
to  the  crank-pin.     To  follow  the  motion  of  the  crank,  there- 
fore, the  cylinder  and  piston  oscillate  on  two  trunnions  pro- 
jecting opposite  the  middle  of  the  former,  the  piston  making 
its  strokes  within  the  cylinder  correspondingly.    F  is  a  fly- 
wheel on  the  main  shaft.     The  under  portion  of  the  casting 
containing  the  cylinder  contains  ports  as  shown  and  terminates 
in  an  accurately  formed  cylindrical  surface  concentric  with 
the  trunnions  on  which  it  is  mounted.     As  the  cylinder  oscil- 
lates, this  surface  moves  in  water-tight  contact  with  a  corre- 
sponding surface  of  the  fixed  base  M,  and  in  such  a  way  that 
each  of  the  two  ports  is  caused  to  communicate  alternately 
with  the  space  S  supplying  the  pressure-water,  and  the  exhaust 
space  E  through  which  the  water  escapes,  after  use,  into  the 
atmosphere.     In  the  position  shown  in  the  figure  the  piston 
is  moving  toward  the  right  and  pressure-water  is  acting  on 
its  left-hand  face;   while  the  water  used  in  the  previous  stroke 
is  now  escaping  through  the  right-hand  port  into  the  exhaust- 
passage  E.     On  the  return  stroke,  the  crank-pin  is  passing 
below  the  main  shaft,  the  right-hand  port  communicates  with 
S,  and  that  on  the  left  with  E.     The  engine  is  therefore  double- 
acting.     It   has   been   cons'derably   used  in  German. y    Two 
engines  of  this  kind  are  sometimes  used,  acting  on  the  same 
shaft  but  with  cranks  90°  apart;    so  that  one  engine  or  the 
other  is  always  in  a  position  to  start. 

153.  Pressure-engine  with  Variable  Stroke. — -When  a  water- 
pressure  engine  is  employed  to  operate  a  hoist,  or  to  turn  a 
capstan,  economy  in  the  use  of   "  pressure- water/'  which  is 
usually  paid  for  by  volume,  is  favored  by  proportioning  the 
amount  of  water  to  the  power  actually  needed  for  raising  the 
load,  which  m^y  be  great  or  small  at  different  times,  or  perhaps 
only  that  of  the  hoisting-chain;   and  this  is  done  in  the  Rigg 


§    154.  WATER-PRESSURE    ENGINES.  251 

and  Hastie  engines  by  an  automatic  change  in  the  length  of 
stroke.  In  the  Rigg  engine,  three  (or  four)  cylinders  radiate 
from  a  common  fixed  shaft  turning  about  it  and  also  oscillating 
somewhat,  their  pistons  being  single-acting  and  thrusting  out- 
wardly against  points  in  the  rim  of  an  encircling  ring  (or  "  fly- 
wheel") secured  upon  the  revolving  shaft  of  the  capstan  or 
hoist.  This  revolving  shaft  is  parallel  to  the  first  shaft,  but 
eccentrically  placed  with  regard  to  it.  The  length  of  stroke 
of  each  piston  is  equal  to  twice  the  distance  between  the  axes 
of  the  two  shafts.  A  governor  is  so  connected  with  a  hydraulic 
11  relay"  motor  that,  any  slight  change  of  speed  due  to  the 
power  exerted  by  the  pistons  being  a  little  in  excess  or  deficiency 
of  what  it  should  be  for  the  proper  constant  speed  in  overcoming 
the  resistance  offered  (whatever  its  amount),  the  distance 
between  the  two  shafts  (and  consequently  the  length  of  stroke) 
is  altered,  until  the  speed  returns  to  the  proper  value.  (See 
Elaine's  Hydraulic  Machinery,  p.  257.)  By  this  device  the 
amount  of  pressure-water  used  is  made  nearly  proportional  to 
the  power  actually  required. 

154.  Piston  with  Large  Rod.     Economy  of  Water. — In  the 
cylinder  of  a  hydraulic  crane  using  pressure-water  the  follow- 
ing device  is  sometimes  adopted  for  economy  in  the  use  of  the 
water.     When  the   piston  P  in  the 
cylinder  C  (see  Fig.  118)    makes    a 
full  stroke  from  right  to  left,  the  load 
on  the  crane  is  lifted  through  its  full 
range.     The  diameter  of  the  piston- 
rod  is  so  great  that  the  annular  area 
forming  the  left  face  of  the  piston  is 
only  about  one-half  that  of  the  right- 
hand  face   (full  circle).    If  a  heavy 

load  is  to  be  lifted,  pressure-water  is  admitted  on  the  right,  at  cy 
while  the  other  end  of  the  cylinder  e  (already  filled  with  water) 
is  put  into  communication  with  the  "  tail-water"  or  exhaust. 
The  resulting  working  force  is  then  a  maximum  and  the 

amount  of  pressure-water  used  is  jd2-l  cub.  ft.;  where  /  is  the 


252 


HYDRAULIC    MOTORS. 


§155. 


length  of  stroke  and  d  the  diameter  of  cylinder.  But  if  a  light 
load  is  to  be  lifted,  both  sides  of  the  piston  are  thrown  into 
communication  with  the  supply  of  pressure-water;  the  resultant 
working  force  has  now  only  half  its  former  value  and  (since 
the  water  previously  in  e  is  forced  back  into  the  pressure- 
pipes)  only  half  the  amount  of  pressure-water  is  used  in  the 
stroke. 

155.  Hydraulic  Accumulators.  —  Many  of  the  smaller  ma- 
chines used  in  manufacturing  plants,  dock-yards,  etc.,  such  as 
riveters,  presses,  punching-  and  shearing-machines,  cranes,  etc., 
etc.,  require  for  their  operation  a  store  of  fluid  under  high 
pressure;  their  action  being  usually  intermittent.  Both  com- 
pressed air  and  water  are  employed  as  fluids  for  this  purpose, 
the  former  being  extensively  used  in  America,  while  the 
latter  is  given  the  preference  in  England  and  the  continent  of 
Europe. 

As  natural  reservoirs  rarely  provide  heads  of  more  than 
300  or  400  ft.  (or  hydrostatic  pressures  of  more  than  130  to 
170  Ibs.  per  sq.  in.),  artificial  means  must 
in  many  instances  be  adopted  for  creat- 
ing and  maintaining  higher  pressures, 
up  to  2000  Ibs.  per  sq.  in.,  in  a  confined 
body  of  fluid.  When  water  is  used, 
the  storage  vessel,  etc.,  is  called  a 
hydraulic  accumulator*  Fig.  119  shows 
(in  a  diagrammatic  way)  the  vertical 
section  of  a  simple  design  of  hydraulic 
accumulator.  CD  is  a  strong  cylin- 
drical vessel  into  whose  upper  end 
protrudes  a  ram,  or  plunger,  (i.e.,  pis- 
ton and  rod  in  one,)  AB.  This  plunger 
is  loaded  (with  rings  of  cast  iron,  for 


through  the  pipe  E  until  the  ram  is 
FIG.  119.  raised  to  its   highest  point.     At   e  is 

shown  an  annular  space  or  " gland"  into  which  hemp  packing 

*  See  p.  375  of  the  Engineering  Record  of  Mar.  23,  1907,  for  a  description 
of  the  accumulators  used  in  a  New  York  elevator  equipment. 


§  156.  WATER-PRESSURE   MACHINERY.  253 

is  compressed,  by  the  screws  holding  down  the  cover  of  the 
gland,  to  prevent  leakage  of  water  around  the  ram  (leather 
packing  may  also  be  used;  see  §  157). 

The  pressure-water,  when  needed  for  occasional  operation 
of  machines,  passes  out  through  the  small  pipe  F  and  through 
other  pipes  to  the  particular  machine  needing  to  be  driven, 
and  the  ram  and  its  load  gradually  sink.  When  the  ram 
reaches  a  definite  point  the  pumps  are  started  and  it  is  again 
raised  to  its  highest  position,  when  the  pumps  are  stopped. 
Both  the  starting  and  stopping  of  the  pumps  are  automatic.  If 
the  demand  for  power  is  large  and  fairly  regular,  the  pumps 
may  be  in  action  continuously;  in  which  case  the  loaded  ram 
remains  nearly  stationary,  no  longer  serving  as  a  storehouse  of 
energy,  but  only  as  a  regulator  of  the  pressure. 

If  the  total  load  on  the  ram  is  G  Ibs.  and  it  is  either  stationary 
or  moving  with  a  slow  uniform  velocity,  we  have  for  the  unit- 
pressure  hi  the  confined  water  (neglecting  friction  at  the  gland) 

p  =  (7-M  -d2}  (above  the  atmosphere);  in  which  d  is  the  diam- 

eter of  the  ram  at  the  gland,  e. 

As  to  the  friction  occasioned  by  the  hemp  packing,  which 
has  to  be  highly  compressed  to  prevent  leakage,  it  is  said 
(Elaine)  that  if  the  hemp  is  well  lubricated  this  friction  is 
about 

F    pd.  fff\ 

F=To>    •  v  •  .....   (8) 


in  which  F  will  be  obtained  in  Ibs.,  if  p  is  expressed  in  Ibs.  per 
sq.  in.  and  d  in  inches.  But  in  ordinary  cases  the  value  of 
the  friction  is  quite  uncertain.  From  eq.  (8)  we  should  have 
for  a  5-inch  ram  F  =2.5  per  cent,  of  the  load  G;  which  is  about 
three  times  that  of  a  "U  leather  "  packing  (see  §  156). 

156.  The  Hydraulic  Lift.  Hydraulic  Jack.  Bramah  Press. 
—  Considered  as  a  mere  diagram,  Fig.  119  also  serves  to  illustrate 
the  principle  of  action  of  the  direct-acting  hydraulic  lift*  or 
elevator;  AB  being  a  long  ram  carrying  a  car  at  the  upper 
end.  In  the  actual  construction,  of  course,  the  car  and  ram 


254 


HYDRAULIC   MOTORS. 


§  157. 


are  made  as  light  as  possible,  the  permanent  load  being  largely 
counterpoised  by  weights  attached  to  chains  or  cables  running 
over  pulleys,  so  that  the  fluid  pressure  needed  is  mainly  that 
required  for  the  temporary  load  (passengers,  freight,  etc.). 

The  hydraulic  jack  is  the  same  in  principle,  and  also  the 
Bramah  hydraulic  press,  the  load  to  be  lifted,  or  pressure  to 
be  applied  (to  a  bale  of  cotton,  for  instance),  playing  the  part 
of  the  loads  on  the  ram  in  Fig.  119.  The  hydraulic  jack  usually 
contains  its  own  supply  of  liquid  (oil  or  water)  in  a  special 
reservoir,  a  hand-pump,  on  the  side  of  the  apparatus,  being 
used  to  pump  it  into  the  space  under  the  plunger. 

157.  The  Differential  Accumulator.  Leather-packing.  In- 
tensifier.  —  In  Tweddell's  differential  accumulator  (Fig.  120)  we 
have  an  inversion  of  position.  The  ram  AB  (dense  black 

shading)  is  fixed  and  placed  below, 
and  the  cylindrical  vessel,  CD,  which 
is  loaded  and  movable,  is  placed 
above.  Also,  the  ram  protrudes 
through  the  loaded  vessel  at  A, 
requiring  two  packings  for  the  two 
sliding  joints.  Here  the  diameter  df 
of  the  upper  portion  of  the  ram  is 
a  little  smaller  than  that,  d,  of  the 
lower  portion.  Consequently  the 
whole  weight,  G  Ibs.,  of  the  cylinder 
CD  and  the  loads  upon  it,  is  borne 
by  the  upward  fluid  pressure  on 
the  area  of  the  ring  (difference  of 
the  areas  of  the  two  circles) 

•gd2—jd'2;    i.e.,    the  unit    pressure 

(above  the  atmosphere)  in  the  space 
FIG.  120.  e>  when  the  whole  load  is  sustained,  is 


if  friction  is  neglected.    By  this  device,  called  a  "differential 


§  157.  HYDRAULIC    ACCUMULATORS.  255 

accumulator"  less  load  is  needed  to  produce  a  given  fluid  pres- 
sure, but  the  amount  of  pressure- water  leaving  the  vessel  for 
each  foot  descent  of  the  loaded  cylinder  is  quite  small;  thus 
necessitating  very  frequent  working  of  the  pumps,  or  perhaps 
their  continuous  action,  to  supply  the  requisite  amount. 

Water  is  pumped  into  the  interior  space  e  through  the  small 
pipe  F  (which  also  serves  as  an  outlet  if  the  flow  is  the  other 
way)  and  through  passageways  in  tr^e  ram  itself,  as  shown. 
The  two  recesses  or  glands  are  furnished  with  water-tight 
-"U-leather"  packings  like  the  one  shown  at  A  in  Fig.  121. 


B 

FIG.  121. 

This  is  made  of  a  single  piece  of  leather,  pressed  into  shape 
(after  softening  by  hot  water)  in  a  proper  mould.  The  con- 
cave side  is  turned  toward  the  interior  of  the  cylinder  and  thus 
exposes  that  side  to  the  high  internal  pressure.  This  pressure 
keeps  the  leather  pressed  tightly  both  against  the  surface  of 
the  ram  and  that  of  the  gland  cavity,  thus  providing  a  water- 
tight joint.  For  the  best  results  the  outside  of  the  ram  should 
be  of  gun-metal  or  copper,  as  also  the  lining  of  the  gland  cavity. 
For  small  rams  or  for  piston-rods  the  "  hat  -leather"  packing 
(see  B  in  Fig.  121)  serves  a  similar  purpose.  (This  cut  is 
from  the  advertisement  of  the  Detroit  Leather  Specialty  Co. 
of  Detroit,  Mich.) 

Leather  packings  are  more  expensive  than  those  of  hemp, 
but  offer  much  less  friction.  According  to  the  experiments  of 
Mr.  Hick,  the  friction  offered  by  a  well-lubricated  U-leather 
packing  is 

F =0.032pd; (10) 

where  p  is  the  internal  fluid  pressure  in  Ibs.  per  sq.  in.,  and  d 
the  diameter  of  the  ram  in  inches;  F  being  then  obtained  in 


FIG.  122. 


w 


FIG.  123. 


FIG.  124. 


256 


I  158.  THE   HYDRAULIC   RAM.  257 

Ibs.     For  a  new  leather,  or  one  badly  lubricated,  the  numeral 
should  be  0.047. 

If  the  load  on  the  ram  in  Fig.  119  is  replaced  by  the  down- 
ward hydrostatic  pressure  on  a  piston  of  much  larger  diameter 
than  that  of  the  ram,  and  capable  of  vertical  motion  in  a 
fixed  cylinder  to  which  water  from  a  comparatively  low  source 
is  admitted,  this  piston  being  attached  to  the  upper  end  of 
the  ram,  the  device  is  called  an  intensifier.  In  such  a  case, 
no  water  need  be  withdrawn  from,  nor  pumped  into,  the  upper 
cylinder. 

158.  The  Hydraulic  Ram  is  a  combined  water-motor  and 
pump  working  in  successive  pulsations  and  by  a  kind  of  mild 
"water-hammer"  action.  This  action  depends  on  the  inter- 
mittent starting  and  stopping  of  the  cylinder  of  water  in  the 
supply-pipe  (or  "  drive-pipe/'  as  it  is  called  in  this  connection). 
In  the  simpler  forms  the  machine  consists  of  an  air-vessel,  W 
(see  Fig.  122);  of  two  valves,  the  " waste- valve"  and  the 
"  check- valve ";  and  of  two  pipes,  viz.,  BA,  the  drive-pipe, 
supplying  water  from  the  supply-pond  S,  and  the  delivery- 
pipe  DF,  through  which  a  certain  amount  of  the  water  is 
pumped  into  the  receiving- tank  or  reservoir,  R,  at  a  higher 
elevation  than  the  supply-pond.  H  is  the  net  head  through 
which  water  is  raised  (the  "lift"),  and  h  the  working-head 
(or  "fall")  of  the  " waste- water "  (or  "motive  water"),  or 
that  which  escapes  through  the  waste-valve  at  F.  If  Q  (cub. 
ft.  per  hour,  say)  is  the  rate  at  which  water  is  pumped  through 
the  pipe  F}  and  q  the  volume  per  hour  of  flow  through  the 
waste- valve,  then  the  rate  of  flow  through  the  supply  drive- 
pipe  BA  is  Q+q;  see  figure. 

Fig.  123  gives  a  (diagrammatic)  vertical  section  of  the 
ram  proper.  At  the  beginning  of  a  cycle,  or  pulsation,  the 
waste-valve  E  at  the  lower  extremity  of  the  drive-pipe  is  open 
and  the  check-valve  V  at  the  base  of  the  air-vessel,  TF,  is  shut, 
being  held  shut  for  the  time  being  by  the  high  pressure  in  the 
air-vessel,  which  communicates  by  pipe  DF  with  upper  tank 
R.  M  is  a  confined  body  of  compressed  air.  Under  the 
action  of  gravity  the  water  in  the  drive-pipe  begins  to  move 

17 


258  HYDRAULIC    MOTORS.  §  158. 

and  flow  out  into  the  air  at  F,  acquiring  greater  and  greater 
velocity  (unsteady  flow).  This  velocity  is  finally  so  great  that 
the  pressure  of  the  current  on  the  under  side  of  valve  E  becomes 
sufficient  to  close  it  abruptly.  The  moving  body  of  water 
in  pipe  CBA  is  now  slightly  checked  in  its  velocity  and  becomes 
compressed  until  the  pressure  rises  (very  quickly)  to  a  value 
a  little  in  excess  of  that  on  the  upper  side  of  valve  F,  when 
this  latter  valve  opens  and  a  portion  of  the  "  drive- water "  is 
forced  into  the  base  of  the  air-vessel,  until  its  kinetic  energy 
and  velocity  are  entirely  exhausted;  the  immediate  effect 
being  a  rising  of  the  water-surface  in  the  air-vessel  and  a  further 
compression  of  the  confined  air  in  M.  The  water  in  the  drive- 
pipe  having  thus  been  brought  to  rest  but  being  still  slightly 
compressed,  an  elastic  recoil  or  rebound  takes  place  and  the 
pressure  in  the  spaces  e  and  v  quickly  falls  to  a  low  value, 
Jess  than  one  atmosphere,  so  that  the  valve  V  closes;  while 
E  is  opened,  both  on  account  of  its  weight  and  of  the  pressure 
of  the  outer  air.  In  other  words,  the  kinetic  energy  possessed 
by  the  drive- water  when  the  valve  E  first  closes  is  expended 
In  compressing  itself  (slightly)  and  then  (mainly)  in  compressing 
the  air  in  the  air-vessel  into  a  smaller  compass.  Another 
cycle  now  begins;  and  so  on,  indefinitely.  While  the  water 
in  the  drive-pipe  is  gradually  acquiring  velocity  in  the  next 
cycle  (and  this  occupies  much  the  greater  part  of  the  time  of 
a  cycle)  the  compressed  air  in  M  expands  and  recovers  its 
former  volume,  forcing  some  of  the  water  in  the  base  of  the 
air-vessel  through  the  pipe  DF  into  the  tank  R;  in  fact,  the 
flow  in  this  pipe  is  fairly  continuous  and  "steady,"  if  a  large 
air-vessel  is  provided. 

A  small  valve  not  yet  described,  and  not  shown  in  the 
figure,  is  the  "snifting-valve,"  in  the  side  wall  of  the  space  e, 
opening  inward  and  closing  the  entrance  of  a  small  pipe  leading 
to  the  outer  air.  At  the  time  of  the  recoil  a  small  quantity 
of  ah*  enters  and  a  little  later  is  carried  into  the  air-vessel; 
to  make  good  the  air  lost  by  being  dissolved  in  the  (high-pres- 
sure) water  in  the  air-vessel. 

In  Fig.  124  is  shown  a  "No.  6"  ram  made  by  the  Goulds 


§  159.  THE    HYDRAULIC    RAM.  259 

Manufacturing  Co.  of  Seneca  Falls,  N.  Y.  The  waste-valve 
is  seen  on  the  right,  while  the  opening  for  attachment  of  delivery- 
pipe  appears  directly  in  the  front,  a  little  to  the  left  of  the  base 
of  air-vessel.  The  long  horizontal  chamber  below  forms  a  con- 
tinuation of  the  drive-pipe  which  is  attached  to  it  at  the  extreme 
left-hand  lower  corner  of  the  figure.  The  drive-pipe  intended 
for  use  with  this  size  of  ram  is  2£  in.  in  diameter  and  of  a  length, 
/,  equal  to  that  of  lift  and  fall  combined;  that  is,  l=H+h, 
while  the  delivery-pipe  has  a  diameter  of  1J  inches. 

159.  Hydraulic  Ram.  Efficiency.  Experiments.  —  If  we  con- 
sider the  useful  power  obtained  to  be  the  raising  of  Qf  Ibs. 
of  water  each  hour  from  the  level  of  S  to  that  of  R  (see  Fig.  122)  ; 
that  is,  through  a  height  H-}  and  that  the  whole  power  ex- 
pended is  that  due  to  a  weight  qy  of  water  (from  waste-valve) 
acting  each  hour  through  a  height  h;  we  obtain  the  Rankine 
form  for  the  efficiency,  viz., 

QH 


The  machine,  however,  is  a  complex  one;  and  if  we  take 
d'Aubuisson's  view  that  the  energy  received  per  unit  of  time 
by  the  ram  is  (Q+q)fh,  and  that  the  useful  effect  obtained 
therefrom  is  the  raising  of  Qf  Ibs.  of  water  per  unit  of  time 
through  a  height  H  +  h,  the  form  for  the  efficiency  becomes 

Q(H-\-h) 


The  Rankine  form  is  the  one  more  generally  employed  and 
will  be  adopted  here.  Under  ordinary  circumstances  (H  large 
compared  with  h)  results  based  on  (2)  are  not  largely  in  excess 
of  those  obtained  from  (1). 

From  extensive  experiments  made  by  himself  in  1804 
Eytelwein  recommends  the  following  relations  to  be  adopted 
for  the  best  results: 

If  Q  and  q  are  expressed  in  cub.  ft.  per  sec.,  the  diameter 
of  the  drive-pipe  should  have  a  value 

d=[\/1.63(Q  +  g)]  feet  ......     (3) 


260  HYDRAULIC    MOTORS.  §  159. 

The  length  I  of  the  drive-pipe  should  be 

Z=#  +  A-K2ft.)X(#-A) (4) 

The  volume  of  the  air-vessel  should  be  equal  to  that  of  the 
delivery-pipe,  and  the  diameter  of  the  latter  should  be  about 
one  half  of  that  of  the  drive-pipe.  The  opening  of  the  waste- 
valve  should  have  the  same  sectional  area  as  the  drive-pipe, 
and  the  weight  of  this  valve  should  be  as  small  as  possible. 
The  drive-pipe  should  be  as  straight  and  free  from  friction  as 
practicable. 

For  the  best  results  the  length  of  stroke  made  by  the  waste- 
valve  in  closing  should  not  be  too  great. 

With  these  proportions  adopted,  Eytelwein  found  that  the 
efficiency  (Rankine  form)  diminished  with  an  increase  of  the 
ratio  of  the  lift  H  to  fall  h;  nearly  according  to  the  relation 
(y  denoting  the  efficiency) 

-i)  =  1.12  -0.2V  (H+h)       ....     (4a) 

for  a  range  of  values  of  the  ratio  H  +h  from  1  to  20. 
This  gives  the  following  table : 

For#-a=    1  2  4  6          10         15         20 

77  =  0.92      0.84      0.72      0.63      0.49      0.34      0.23 

A  few  experiments  were  made  by  the  present  writer  on  a 
small  ram  (No.  2  Goulds)  at  Cornell  University*  in  March 
1899.  The  data  and  results  are  tabulated  below.  The  drive- 
pipe  had  a  diameter  of  £  in.  and  was  51  ft.  long.  The  delivery- 
pipe  was  one  inch  in  diameter  and  comparatively  short,  offering 
but  little  loss  of  head.  Column  A  gives  pulsations  per  minute, 
and  B  the  height  of  stroke  of  the  waste-valve  in  closing.  Qf 
and  qr  are  Ibs.  per  minute ;  H  and  h  are  feet. 

These  experiments  were  repeated  to  insure  accuracy,  each 
of  the  three  horizontal  lines  giving  the  mean  results  of 
several  in  close  agreement  with  each  other.  In  No.  1  the 
full  weight  of  the  waste-valve,  which  was  6f  ounces,  was  opera- 

*  See  also  Pi  of.  R.  C.  Carpenter's  experiments,  mentioned  in  Kent's 
Mechanical  Engineers  Pocket-book. 


§  160. 


THE   HYDRAULIC    RAM. 


261 


Experiments  on  a  Hydraulic  Ram,  Cornell  University, 
March  1899. 


1 

Whole 

No. 

Time; 
Min. 

A 

B 
In. 

Qr 

H 

?r 

h 

EfL. 

1 

12 

80 

i 

5.50 

49.4 

26.7 

17 

0.598 

2(8) 

15 

66 

7 
8" 

2.30 

56.4 

21.8 

10 

0.595 

3(8) 

16.7 

98 

I 

4.82 

49.4 

20.6 

17 

0  680 

tive;  there  being  no  provision  to  counterpoise  a  portion  of  it; 
and  the  length  of  its  movement,  or  "stroke,"  was  J  in.  In 
the  other  two,  Nos.  2(s)  and  3(s),  a  light  spring  was  employed 
by  the  use  of  which  the  waste-valve  was  virtually  relieved 
of  about  one-half  of  its  weight  (though,  of  course,  its  "mass" 
was  practically  unchanged).  In  No.  2(s),  the  length  of  stroke 
of  valve  being  the  same  as  before,  the  efficiency  is  maintained 
at  nearly  60  per  cent.,  notwithstanding  the  fact  that  the  ratio 
H +h  is  nearly  double  what  it  was  in  No.  1.  This  is  doubt- 
less due  to  the  (relatively)  quick  closing  of  the  valve.  In 
No.  3(s)  the  stroke  has  been  made  shorter  with  the  effect  of 
increasing  the  efficiency  to  68  per  cent.;  to  which  the  decrease 
in  the  ratio  H  +h  has  also  probably  contributed  somewhat. 
The  pulsation  is  here  very  rapid:  98  to  the  minute. 

With  a  somewhat  longer  drive-pipe  (say  65  ft.)  the  efficiency 
would  probably  have  been  higher. 

160.  Hydraulic  Ram.  Special  Designs. — In  some  designs 
the  check-valve  shown  as  V  in  Fig.  123  is  placed  on  the  side 
of  the  chamber  e,  the  top  space  in  which  is  then  used  as  a  dome, 
or  pocket,  in  which  to  entrap  permanently  a  small  body  of 
air,  which  serves  to  lighten  the  shock  and  water-hammer  effect, 
preventing  the  pressure  in  e  from  rising  much  above  that  in 
the  air-vessel  at  any  time. 

The  Rife  "Hydraulic  Engine"  (see  Engineering  News  of 
Dec.  31, 1896)  is  a  large  hydraulic  ram  in  which  the  effective 
weight  of  a  waste-valve  can  be  varied  by  means  of  a  weight 
sliding  on  a  lever.  It  can  also  be  arranged  so  that  the  water 


262  HYDRAULIC   MOTORS.  §  160. 

pumped  may  be  taken  from  a  different  source  (purer  water, 
for  instance)  from  that  of  the  drive-water;  the  periodic  recoil 
of  the  drive-water  serving  to  cause  the  entrance  ("by  suction") 
into  the  space  e,  of  each  new  instalment  of  the  water  to  be 
pumped.  A  description  of  some  interesting  tests  of  a  Rife 
hydraulic  ram  may  be  found  in  the  Stevens  Indicator  of  April 
1898  (published  at  the  Stevens  Institute,  Hoboken,  N.  J.). 
The  highest  efficiency  obtained  was  75.6  per  cent.* 

A  large  ram  designed  by  Prof.  D.  W.  Mead  of  Chicago  is 
in  operation  at  the  village  of  West  Dundee,  Illinois,  in  con- 
nection with  the  local  water-works.  This  machine,  with  a 
drive-pipe  2200  ft.  long  and  10  inches  in  diameter,  under  a 
head  of  55  ft.,  delivers  water  into  a  stand-pipe  115  ft.  above 
the  ram.  On  account  of  the  great  length  of  this  drive-pipe, 
the  waste-valve,  which  is  circular  and  8  inches  in  diameter, 
was  given  a  lift,  or  "  stroke/'  of  only  J  inch,  thus  giving  an 
area  of  discharge  equal  to  only  about  one-twelfth  of  the  sec- 
tional area  of  the  drive-pipe;  and  the  aggregate  area  of  the 
(nine)  check-valves  at  the  base  of  the  air-vessel  is  greater  than 
the  sectional  area  of  the  10-inch  drive-pipe.  The  duration  of 
one  cycle  or  pulsation  of  this  ram  is  4J  seconds,  and  the  pressure 
in  the  space  e  (Fig.  123)  never  exceeds  by  more  than  2J  Ibs. 
the  pressure  in  the  air-vessel,  which  is  50  Ibs.  per  sq.  in.  (above 
atmos.) .  According  to  the  indicator-cards  taken,  this  highest 
pressure  endures  only  about  one  second. 

An  account  of  the  Pearsall  Hydraulic  Engine,  and  of  a  test 
of  the  same,  may  be  found  in  the  Engineering  News  for  Sept.  28, 
1889.  This  is  a  large  hydraulic  ram  in  which  the  waste-valve 
is  cylindrical  in  form  and  is  closed,  not  violently,  by  the  current 
of  flowing  water,  but  quietly,  by  the  action  of  a  small  motor 
worked  by  compressed  air  taken  from  the  air-vessel.  The 
essential  principle  involved  (referring  now  to  Fig.  123)  is  prac- 
tically stated  by  saying  that  the  chamber  e  is  so  furnished  with 
valves  (besides  V)  admitting  ah-  from  the  outside  at  the  proper 

*  In  Engineering  News  of  Aug.  3,  1905,  p.  127,  is  an  account  of  two 
"Foster  Impact  Engines,"  or  large  hydraulic  rams,  installed  at  Bradford, 
R.  I.,  and  working  with  high  efficiency. 


§  161.  HYDRAULIC    RAMS.  263 

times  that  sufficient  air  is  entrapped  and  cushioned  under 
the  valve  V  and  finally  forced  into  the  air-vessel  along  with 
the  water  pumped,  not  only  to  make  good  the  ordinary  losses 
of  air  in  the  air-vessel,  but  also  to  furnish  what  is  needed  for 
the  operation  of  the  small  motor  which  opens  and  closes  the 
waste-valve.  In  this  way  the  efficiency  is  increased;  71  per 
cent,  having  been  attained  in  the  case  referred  to.  From  the 
data  given  it  appears  that  H  was  83  ft.;  and  A,  17  ft.;  while 
the  diameter  of  drive-pipe  was  12  in. 

161.  The  Phillips  Hydraulic  Ram  is  made  in  a  great  variety 
of  sizes  (from  3  in.  to  48  in.  diameter  of  drive-pipe)  by  the- 
Columbia  Engineering  Works  of  Portland,  Oregon,  U.  S.  A. 
Like  the  Pearsall  ram  it  has  a  cylindrical  waste-valve,  the* 
closing  of  which  is  brought  about  without  violence,  but  the 
motor  which  operates  its  opening  and  closing  is  a  small, 
single-acting,  water-pressure  engine,  receiving  its  pressure- 
water  from  the  supply -pond.  This  takes  but  little  power; 
as  the  waste-valve  from  its  cylindrical  form  is  always  "bal- 
anced." Sufficient  air  is  cushioned  and  entrapped  to  supply 
the  small  losses  of  the  air-vessel  and  also,  by  a  small  piston, 
to  operate  the  exhaust-valve  of  the  pressure-engine. 

The  diagram  shown  as  Fig.  125  has  been  kindly  furnished 
by  the  makers  of  the  ram,  and  the  following  clear  description 
of  its  operation  is  taken  from  their  pamphlet: 

"The  drive-pipe  is  connected  at  A.  This  pipe,  which  is 
of  a  size  suitable  for  handling  the  desired  quantity  of  water, 
determines  the  normal  size  of  the  ram.  The  waste-valve  Q 
being  open,  the  water  flowing  down  the  drive-pipe  escapes. 
At  the  same  time  water  from  the  main  supply  is  flowing 
through  the  operating-pipe  G  and  the  upper  part  of  operating- 
valve  D  into  the  Waste-valve  cylinder  H.  This  water  raises  the 
waste-valve  piston  E  and  with  it  the  waste- valve  C,  thus 
closing  the  opening  and  causing  a  stoppage  in  the  main  flow 
of  the  water.  The  energy  stored  up  in  the  flowing  water  is 
now  liberated  and  forces  part  of  the  latter  through  discharge- 
valves  LL  into  the  main  air-chamber.  This  discharge  is  con- 
tinued until  an  equilibrium  between  the  respective  pressures 


264  HYDRAULIC    MOTORS.  §   162. 

above  and  below  the  discharge-valve  is  established.  At  the 
time  the  waste-valve  closes,  air  is  lodged  in  the  chamber  /. 
This  air  is  compressed  by  the  moving  water  column  and  part 
of  it  is  delivered  through  pipe  M  into  the  air-chamber,  thus 
giving  the  latter  a  constant  supply  at  each  impulse.  The 
pressure  in  chamber  /  will  also  be  imparted  through  pipe  K 
to  the  lower  part  of  the  operating-valve  D,  where  it  raises  the 
piston  N,  thereby  closing  the  smaller  beveled  valve  above  and 
cutting  off  communication  between  the  operating-pipe  G  and 
the  under  side  of  waste-valve  piston  E.  At  the  same  time,  an 
exhaust  opening  is  made  at  the  top  of  operating-valve  D  for 
the  operating  water,  which  now  escapes  and  allows  piston  E 
and  with  its  waste-valve  0  to  drop  by  gravity,  thus  making  the 
beginning  of  a  new  cycle." 

162.  Hydraulic  Air-compression. — Intermittent. — The  action 
of  the  hydraulic  ram  is  easily  arranged  to  bring  about  the 
compression  of  a  confined  body  of  air,  as  also  its  delivery  into 
a  storage-tank,  without  the  pumping  of  any  water.  The 
apparatus  used  at  the  Mt.  Cenis  tunnel  is  described  in  Davey's 
"Pumping  Machinery"  (London,  1900),  p.  285.  At  the 
bottom  of  a  closed  vertical  cylindrical  vessel  containing  air  at 
one  atmosphere  pressure,  water  from  a  pipe  communicating 
with  an  elevated  reservoir  is  permitted  to  enter  by  the  opening 
of  a  valve;  its  -initial  velocity  being  zero.  The  resistance 
offered  by  the  air  to  the  advance  of  the  water  into  the  vessel 
being  small  at  the  beginning,  the  velocity  of  the  water  increases 
at  first  and  reaches  a  maximum,  after  which  its  energy  of  motion 
is  gradually  given  out  in  compressing  the  air  still  further,  and, 
later,  when  the  air-pressure  is  sufficient  to  open  the  valve 
leading  to  the  storage- tank,  to  perform  the  work  of  delivery; 
that  is,  to  force  the  ah1  at  this  final  constant  pressure  into  the 
tank.  Dimensions  are  so  designed  that  the  water  is  brought 
to  rest  just  before  it  reaches  the  upper  end  of  the  compressing 
vessel.  At  that  instant,  by  the  automatic  operation  of  the 
proper  valves,  further  entrance  of  water  is  prevented  and  that 
already  in  the  compressing  vessel  allowed  to  flow  out  into  the. 
atmosphere  and  the  vessel  to  fill  up  with  a  new  charge  of  air 


I 

•s 


s 


265 


266  HYDRAULIC   MOTORS.  §  162. 

from  the  outside;  and  the  cycle  is  repeated  indefinitely.  In 
this  way  air  may  be  compressed  to  a  pressure  very  much  greater 
than  the  hydrostatic  pressure  corresponding  to  the  head  of 
water  used.  At  the  Mt.  Cenis  tunnel  the  head,  h,  of  the  supply- 
pond  was  85  ft.,  and  the  final  pressure  of  the  air  75  Ibs.  per 
sq.  in.  (above  atmos.).  The  theory  of  this  operation  is  as 
follows: 

Let  the  sectional  area  of  the  vertical  compressor  cylinder 
be  Ff  and  its  length  Z',  and  let  the  design  be  such  that  the  water 
which  has  entered  it  during  the  compressing  of  the  air  com- 
pletely fills  it,  and  has  just  come  to  rest  at  the  end  of  the  stroke. 
The  weight  of  this  water  is  then  F'l'f.  Also,  let  h'  denote 
the  vertical  depth  of  the  center  of  gravity  below  the  surface 
of  the  supply-pond.  Let  pm  indicate  the  (unit)  pressure  of 
the  compressed  air  in  the  storage-tank,  and  pa  that  of  the 
outer  atmosphere. 

We  are  now  to  note  that  W,  the  work  of  overcoming  the 
air-resistance  at  the  front  face  of  the  advancing  body  of  water 
during  the  stroke,  will  be  the  same  (if  we  assume  the  com- 
pression to  be  adiabatic)  as  that  on  the  front  face  of  the  piston 
of  the  air-compressor  treated  on  p.  636,  M.  of  E.  If,  in  the 
analysis  of  pp.  631  to  637,  M.  of  E.,  the  more  accurate  value 
1.41,  of  Poisson's  exponent  had  been  used,*  instead  of  1.50 
(see  p.  623,  M.  of  E.),  we  should  have  obtained  for  the  work 
done  in  one  stroke  by  the  thrust  in  the  piston-roc?  of  the  air- 
compressor  of  pp.  636  and  637  (after  a  little  transformation 
and  using  present  notation)  the  expression 

0.290         n 

(5) 


in  place  of  eq.  (2)  of  p.  637,  M.  of  E. 

Now  consider  the  collection  of  rigid  bodies  comprising  all 
the  particles  of  water  in  the  pond  and  supply-pipe  (long  or 
short),  and  the  fact  that  all  these  particles  are  at  rest  both  at 
beginning  and  end  of  the  stroke,  so  that  both  initial  and  final- 
amounts  of  kinetic  energy  are  zero.  During  the  stroke  the 
center  of  gravity  of  this  whole  body  of  water,  whose  weight 

*  This  change  has  been  made  in  the  revised  edition  (1908)  of  the  author's 
Mechanics  of  Engineering. 


§    162.  HYDRAULIC   AIR-COMPRESSION.  267 

is  G  Ibs.,  sinks  through  some  small  vertical  distance  Ah,  so 
that  the  working  force  G  does  the  work  G-  Ah\  while  the  surface 
of  the  pond  sinks  slightly,  through  a  distance  As,  so  that  the 
atmospheric  pressure  on  that  surface  does  the  work  F"-pa-4s 
(where  F"  is  the  area  of  pond  surface).  This  last  item  of  work 
corresponds  (and  is  equal)  to  the  work  done  by  the  atmosphere 
on  the  hinder  face  of  the  piston  in  the  ordinary  air-compressor; 
that  is,  we  may  write  F"pa-Js=F'pa-l'  (since  the  volumes 
F"As  and  F'V  are  equal);  and  are  also  to  note  that  W'  =  W 
+F'pJ,',  i.e.,  that  W  =  W+F"pa  -  As.  Hence  the  term  F"pa  -  As 
will  cancel  out  in  the  final  summation  of  work  (see  below). 

By  the  theorem  of  work  and  energy,  then,  (p.  149,  M.  of  E.,) 
applied  for  one  stroke  to  the  collection  of  rigid  bodies  in  ques- 
tion, remembering  that,  by  §  32,  G-  Ah^F'l'f'h',  and  neglect- 
ing all  friction  for  the  time  being,  we  have 


.     .    (6) 
that  is,  finally, 


3-44 


Here  it  is  to  be  noted  that  h'  is  not  the  full  head,  h,  of  the 
"  mill-site,"  but  smaller,  since  the  compressor  cylinder  is  ver- 
tical; and  that  b  denotes  the  height  of  the  water-barometer 
(i.e.,  about  34  ft.  at  sea-level  and  less  at  higher  altitudes). 

Applying  (7)  to  the  case  of  the  Mt.  Cenis  apparatus  assuming 
that  hf  was  85  ft.  and  that  b  was  29.5  ft.,  we  cbtain  the  result 
pm=8.12pa;  and  if  pa  was  12.8  Ibs.  per  sq.  in.,  pm  is  91  Ibs. 
per  sq.  in.  above  the  atmosphere. 

If  in  this  apparatus  the  supply-pipe  is  quite  long  and  of 
length  I,  with  diameter  d  and  sectional  area  F,  and  if  by  allow- 
ing at  first  a  portion  of  the  water  to  escape  into  the  outer  air 
until  that  in  the  pipe  has  a  velocity  =  c  when  permitted  to 
enter  the  compressing  cylinder,  eq.  (6)  becomes 


rfn   \0.290       T 
=3.44^'4(^)        -l],     (8) 


268  HYDRAULIC    MOTORS.  §  163. 

since  we  must  now  introduce  the  initial  kinetic  energy  of  the 
water  in  the  supply-pipe  and  the  work  spent  on  skin  friction. 
(See  eq.  (1)  on  p.  695,  M.  of  E.)  f  is  the  weight  of  unit  volume 
of  water,  and  vm  is  an  average  velocity  of  the  water  in  pipe 
during  the  stroke  (a  value  rather  difficult  to  estimate). 

163.  Hydraulic  Air-compression.  Continuous. — When  water 
is  agitated  in  contact  with  the  atmosphere  it  becomes  charged 
with  small  air-bubbles.  In  still  water  these  bubbles  would 
ascend  with  about  1  ft.  per  second  velocity;  so  that  in  a  de- 
scending current  of  (say)  4  ft.  per  second  they  would  be  carried 
along  by  the  current,  and  with  an  absolute  velocity  of  about 
3  ft.  per  second. 

A  hydraulic  method  for  the  continuous  production  of  com- 
pressed air  is  founded  on  these  facts  and  was  invented  about 
1878  by  Mr.  J.  P.  Frizell.  It  has  the  simplicity  of  involving 
no  moving  parts  whatever. 

It  requires,  of  course,  a  "mill-site77  with  some  fall  h  and 
water-supply  Q  cub.  ft.  per  sec.  A  vertical  "  descending  shaft/' 
or  pipe,  conducts  water  from  the  head-water  to  a  horizontal 
shaft,  which,  again,  leads  into  a  vertical  "  ascending  shaft" 
terminating  under  the  surface  of  the  tail-water.  Water  can 
enter  the  "descending  shaft"  only  by  flowing  over  the  hori- 
zontal edge  of  a  funnel;  the  radial  converging  streams  meet 
each  other  at  the  bottom  where  the  funnel  empties  into  the 
"  descending  shaft  "  and  break  into  foam  by  mutual  collision 
and  agitation  at  that  point;  then  enter  the  top  of  the  de- 
scending shaft.  With  proper  regulation  of  dimensions,  tne 
descending  current,  which  occupies  the  full  section  of  the  shaft, 
has  sufficient  velocity  to  entrain  the  air-bubbles;  which,  after 
the  horizontal  shaft  is  entered,  gradually  work  their  way  to  the 
upper  part  of  this  shaft,  where,  before  the  junction  with  the 
ascending  shaft,  they  rise  and  collect  in  a  bell  or  air-chamber 
and  form  a  body  of  compressed  air  whose  pressure  is  practically 
equal  to  that  (hydrostatic)  corresponding  to  the  depth  of  the 
horizontal  shaft  below  the  surface  of  the  tail-water;  this  supply 
is  drawn  upon  through  proper  air-pipes.  The  water,  having 


§   163.  HYDRAULIC    AIK-COMPRESSION.  269 

left  the  air  behind  it,  rises  through  the  ascending  shaft  and 
joins  the  tail- water  (see  FrizelPs  "Water-power/7  p.  426). 

The  Taylor  method  of  hydraulic  compression  employs  prac- 
tically the  three  shafts  mentioned  above  or  their  equivalent, 
but  the  mode  of  charging  the  water  with  the  air-bubbles  is 
different.  The  water,  in  steady  flow,  enters  the  upper  end 
of  the  descending  shaft  or  pipe  through  a  constricted  sectional 
area;  the  internal  fluid^  pressure  at  that  part  of  the  flow  being 
thus  given  a  value  less  than  one  atmosphere,  by  proper  design 
of  the  parts.  Small  openings  in  the  walls  of  this  constricted 
part  of  the  passageway  communicate  by -pipes  with  the  outer 
atmosphere,  and  through  them  air  is  forced  in  by  the  outside 
air-pressure  and  joins  the  current  of  water  in  the  pipe.  In 
short,  the  principle  of  Sprengel's  air-pump  is  employed  for 
introducing  the  air  (see  p.  656,  M.  of  E.). 

As  before,  the  air-bubbles  become  disengaged  from  the 
water  at  the  lowest  point  of  the  apparatus  where  the  current 
is  slow  and  horizontal,  and  are  collected  in  a  suitable  chamber. 
Of  course,  the  higher  the  pressure  desired  for  the  compressed 
air  the  greater  the  necessary  depth  of  the  lowest  point  of 
the  shafts  below  the  surface  of  the  tail-water. 

Two  installations  involving  the  Taylor  method  have  been 
built:  one  at  Magog,  Province  of  Quebec,  Canada,  where  a 
final  pressure  of  52  Ibs.  per  sq.  in.  is  obtained,  the  efficiency 
of  the  plant  being  about  62  per  cent.;  and  the  other  at  Taft- 
ville,  Conn.  (1900).  The  compressed  air  is  used  to  operate 
compressed-air  engines  either  near  by  or  at  a  distance.  (See 
London  Engineering,  June  1898,  p.  562;  and  FrizelPs  lt  Water- 
power,"  p.  470.  Also  see  Journ.  Assoc.  Engin.  Societies,  Jan. 
1901,  p.  35;  Engineering  News,  May  1901,  p.  406.  For  trans- 
mission of  compressed  air  in  pipes,  see  pp.  786-795,  M.  of  E.) 


APPENDIX 


OF 


DIAGRAMS     AND     TABLES. 


•CONVERSION  SCALES.     (See  p.   190.) 

FRICTION-HEAD  DIAGRAMS  FOR  PIPES.     (See  pp.  189  and  192.) 

DIAGRAMS  FOR  KUTTER'S  COEFFICIENT.     (See  p.  215.) 

FOUR-PLACE  LOGARITHMS. 

FOUR- PLACE  (NATURAL)  TRIGONOMETRIC  RATIOS. 


I 

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8- 


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1-     o. 

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4 

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—  o 


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CO 
(2 

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QN003S    d3d    133J     01900     N  I  6  0  c   3O«  VHOS  I  Q 

6    ooo    oooo    ooo    ooo    oooo    ooo    omo    in  CM  o  <x>    <o  m  *    <o 
o    ooo    OQOO    ooo    omo    mcMOco    coin^t    COCMCM    —  —  — 

§OmO    incMOoo    coin^*    COCMCM    «-»-• 

AO 


18 


QNooag  iJ3d  j.334  oiano  NI 


10          _. 

§^         inCMQOOCDin^t    COCMcMi^i-2 
<o     Tj-focMCM  — ' w »-'  d  d  'd  d   'dddd 


-«a*«S;»?82*5    e.oS^S 

§oooo°oooooo    Ooooooo 
*d  £    *d        'd    "     '    d        *d 


DIAGRAMS. Based  on  the  Formula  of  Ganguillet  and  Kutter. 


SLOPE 

N  FEET  PER  THOUSAND 

1O      0.4      0.1      .025 
1        0.2      .05 


50 


n=.oo9  | 

rt 

«  *=   009  for  well  planed     •£ 
timber  evenly  laid  ** 


SLOPE 
IN  FEET  PER  THOUSAND 

1O      0.4      .10    .025 
1        0.2      .05 


plaster  in  pure  cement  ; 
glazed  surfaces  in  good 
order 


SLOPE 

N  FEET  PER  THOUSAND! 

1O      0.4       .10     .025 
1        0.2      .05 


3      3 
plaster  in  cement  with      »-    ^ 

one-third  sand  .  iron        >»    « 
and  cement     pipes  inf 
good  order  and  well  laid 


250 
240 
230 
220 
210 
200 
190 
180 
170 
160 
150 
140 
130 
120 
110 
100 
90 
•80 
70 
60 
50 

„  I 


DIAGRAMS.  Based  on  the  Formula  of  Ganguillet  and  Kutter. 


230 
220 
210 
200 
190 
180 
170 
160 

140 

130 

120 

110 

100 

90 

80 

70 

60 

50 

40 

c 
£ 
u 

« 

u 


SLOPE 

N  FEET  PER  THOUSAND 

1O      0.4      .10    .025 
1       0.2     .05 


n=.Q12  5 


unplaned  timber  evenly  J~ 


laid  and  continuous.          «     a 


SLOPE 

FEET  PER  THOUSAND 

10      0.4      .10    .025 
1       0.2     .05 


n=.oi3 


ashlar  masonry  and  «  ~. 
well  laid  brick  work  .  *o  -c 
also  the  above  catego  I  ^ 
ries  when  not  in  good  conditioi 
nor  well  laid 


SLOPE 

IN  FEET  PER  THOUSAND 

1O      0.4      .10     .025 
1        0.2     .05 


canvas  lining  on    =     "^ 
frames  ".  brick-work     "C    *o 
ot  tough  iiirtace     foul  X   2i 
irou  pipes  .  badly  jointed 
ceinem 


21O 
200 
190 
180 
170 
160 
150 
140 
130 
120 
110 
100 
90 
80 
70 
60 
50 
40 
30 


DIAGRAMS.  Based  on  the  Formula  of  Ganguillet  and  Kutter. 


200 

190 

180 

170 

160 

150 

140 

130 

120 

110 

100 

90 

80 

70 

60 

50 

40 

30 

20 


SLOPE 

N  FEET  PER  THOUSAND 

1O      0.4      .10     .025 
1        0.2      .05 


11=017 


rubble  in  plaster  or    •& 


cement  in  good  order  ; 
inferior    brick  work  ; 
tubereulated  iron  pipes ; 
very  fine  and    rammed 
gravel. 


For  cylindrical  pipes  or  circular  Sewers,  running  full  or  lial 
"ull.  values  of  R  are  as  follows : 

For  a  diameter  of  12  uicnes~fr  «•  0.25    ft. 


SLOPE 

IN  FEET  PER  THOUSAND 

1O      0.4      .10    .025 
1        0.2     .05 


SLOPE 
IN  FEET  PER  THOUSAND 

1O      0.4     .10     .025 
1       0.2     .05 


canals  in  very  firm  gravel  ;  « 
rubble  in  inferior  condi- 
tion ;  earth  of  even  surface.  ».* 


to 

«       8 
M      6 


=  0.208  ft. 
R  —  o  166  ft. 
R  =  0.125  ft. 
R  «=  o.  104  ft 
R  *  0.083  ft. 


canals  and  rivers  in  perfect  -3. 
order  and  regimen  and  per*  £ 
fectly  fre«  from  stones  and 
weeds.  T 


160 
150 
140 
130 
120 
110 
100 
90 
80 
70 
60 
50 
40 
30 
20 


DIAGRAMS.  Based  on  the  Formula  of  Ganguillet  and  Kutter. 


140 
130 
120 
110 
100 

90 
80 

70 
60 
50 
40 
30 
20 


SLOPE 

N  FEET  PER  THOUSAND 

1O      0.4      .10    .025 
1        0.2     .05 


n=.030  e 


canals  and  rivers  in  earth  'C 

in  moderately  good  order  0$ 

and        regimen,  having  -j~ 
stones  and  weeds 

occasionally  ">. 

K 


SLOPE 

IN  FEET  PER  THOUSAND 

1O      0.4      .10     .025 
1        0.2      .05 


SLOPE 

IN  FEET  PER  THOUSAND 

1O     0.4       .10  .025 
1        0.2.   .05 


n=.035S 


canals  and  rivers  in  bad  J£ 
order  and  regimen, 

overgrown       wjti)  P^ 

vegetation,  and  strewu  .j~ 

with  stones  and  detritus.  ^ 

1 


n=,040£ 


110 
100 
90 
80 
70 
6O 
50 
40 
3O 
20 
10 


LOGARITHMS  (BRIGGS'). 

N 

01234 

56789 

Dif. 

10 

11 
12 
13 
14 

0000  0043  0086  0128  0170 
0414  0453  0492  0531  0569 
0792  0828  0864  0899  0934 
1139  1173  1206  1239  1271 
1461  1492  1523  1553  1584 

0212  0253  0294  0334  0374 
0607  0645  0682  0719  0755 
0969  1004  1038  1072  1106 
1303  1335  1367  1399  1430 
1614  1644  1673  1703  1732 

42 
38 
35 
32 
30 

15 

16 

17 
18 
19 

1761  1790  1818  1847  1875 
2041  2068  2095  2122  2148 
2304  2330  2355  2380  2405 
2553  2577  2601  2625  2648 
2788  2810  2833  2856  2878 

1903  1931  1959  1987  2014 
2175  2201  2227  2253  2279 
2430  2455  2480  2504  2529 
2672  2695  2718  2742  2765 
2900  2923  2945  2967  2989 

28 
26 
25 
24 
22 

2O 

21 

22 
23 
24 

3010  3032  3054  3075  3096 
3222  3243  3263  3284  3304 
3424  3444  3464  3483  3502 
3617  3636  3655  3674  3692 
3802  3820  3838  3856  3874 

3118  3139  3160  3181  3201 
3324  3345  3365  3385  3404 
3522  3541  3560  3579  3598 
3711  3729  3747  3766  3784 
3892  3909  3927  3945  3962 

21 
20 
19 
19 
18 

25 

26 

27 
28 
29 

3979  3997  4014  4031  4048 
4150  4166  4183  4200  4216 
4314  4330  4346  4362  4378 
4472  4487  4502  4518  4533 
4624  4639  4654  4669  4683 

4065  4082  4099  4116  4133 
4232  4249  4265  4281  4298 
4393  4409  4425  4440  4456 
4548  4564  4579  4594  4609 
4698  4713  4728  4742  4757 

17 
16 
16 
15 
15 

3O 

31 
32 
33 
34 

4771  4786  4800  4814  4829 
4914  4928  4942  4955  4969 
5051  5065  5079  5092  5105 
5185  5198  5211  5224  5237 
5315  5328  5340  5353  5366 

4843  4857  4871  4886  4900 
4983  4997  5011  5024  5038 
5119  5132  5145  5159  5172 
5250  5263  5276  5289  5302 
5378  5391  5403  5416  5428 

14 
14 
13 
13 
13 

35 

36 
37 
38 
39 

5441  5453  5465  5478  5490 
5563  5575  5587  5599  5611 
5682-  5694  5705  5717  5729 
5798  5809  5821  5832  5843 
5911  5922  5933  5944  5955 

5502  5514  5527  5539  5551 
5623  5635  5647  5658  5670 
5740  5752  5763  5775  5786 
5855  5866  5877  5888  5899 
5966  5977  5988  5999  6010 

12 
12 

12 
11 
11 

40 

41 
42 
43 

44 

6021  6031  6042  6053  6064 
6128  6138  6149  6160  6170 
6232  6243  6253  6263  6274 
6335  6345  6355  6365  6375 
6435  6444  6454  6464  6474 

6075  6085  6096  6107  6117 
6180  6191  6201  6212  6222 
6284  6294  6304  6314  6325 
6385  6395  6405  6415  6425 
'6484  6493  6503  6513  6522 

11 
10 
10 
10 
10 

45 

46 

47 
48 
49 

6532  6542  6551  6561  6571 
6628  6637  6646  6656  6665 
6721  6730  6739  6749  6758 
6812  6821  6830  6839  6848 
6902  6911  6920  6928  6937 

6580  6590  6599  6609  6618 
6675  6684  6693  6702  6712 
6767  6776  6785  6794  6803 
6857  6866  6875  6884  6893 
6946  6955  6964  6972  6981 

10 
9 
9 
9 
9 

50 

51 
52 
53 

54 

6990  6998  7007  7016  7024 
7076  7084  7093  7101  7110 
7160  7168  7177  7185  7193 
7243  7251  7259  7267  7275 
7324  7332  7340  7348  7356 

7033  7042  7050  7059  7067 
7118  7126  '7135  7143  7152 
7202  7210  7218  7226  7235 
7284  7292  7300  7308  7316 
7364  7372  7380  7388  7396 

9 
9 

8 
8 
8 

N.  B.—  Naperian  log  =  Briggs'  log  x  2.302. 
Base  of  Naperian  system  =  e  =  2.71828. 

LOGARITHMS  (BRIGGS'). 

N 

01234 

56789 

Dif. 

55 

56 
57 
58 
59 

7404  7412  7419  7427  7435 
7482  7490  7497  7505  7513 
7559  7566  7574  7582  7589 
7634  7642  7649  7657  7664 
7709  7716  7723  7731  7738 

7443  7451  7459  7466  7474 
7520  7528  7536  7543  7551 
7597  7604  7612  7619  7627 
7672  7679  7686  7694  7701 
7745  7752  7760  7767  7774 

8 
8 
8 
7 

7 

6O 

61 
62 
63 
64 

7782  7789  7796  7803  7810 
7853  7860  7868  7875  7882 
7924  7931  7938  7945  7952 
7993  8000  8007  8014  8021 
8062  8069  8075  8082  8089 

7818  7825  7832  7839  7846 
7889  7896  7903  7910  7917 
7959  7966  7973  7980  7987 
8028  8035  8041  8048  8055 
8096  8102  8109  8116  8122 

7 

7 
7 
7 
7 

05 

66 
67 
68 
69 

8129  8136  8142  8149  8156 
8195  8202  8209  8215  8222 
8261  8267  8274  8280  8287 
8325  8331  8338  8344  8351 
8388  8395  8401  8407  8414 

8162  8169  8176  8182  8189 
8228  8235  8241  '8248  8254 
8293  8299  8306  8312  8319 
8357  8363  8370  8376  8382 
8420  8426  8432  8439  8445 

7 
7 
6 
6 
6 

70 

71 

72 
73 

74 

8451  8457  8463  8470  8476 
8513  8519  8525  8531  8537 
8573  8579  8585  8591  8597 
8633  8639  8645  8651  8657 
8692  8698  8704  8710  8716 

8482  8488  8494  8500  8506 
8543  8549  8555  8561  8567 
8603  8609  8615  8621  8627 
8663  8669  8675  8681  8686 
8722  8727  8733  8739  8745 

6 
6 
6 
6 
6 

75 

76 

77 
78 
79 

8751  8756  8762  8768  8774 
8808  8814  8820  8825-  8831 
8865  8871  8876  8882  8887 
8921  8927  8932  8938  8943 
8976  8982  8987  8993  8998 

'8779  8785  8791  8797  8802 
8837  8842  8848  8854  8859 
8893  8899  8904  8910  8915 
8949  8954  8960  8965  8971 
9004  9009  9015  9020  9025 

6 
6 
6 
6 
5 

8O 

81 

82 
83 

84 

9031  9036  9042  9047  9053 
9085  9090  9096  9101  9106 
9138  9143  9149  9154  9159 
9191  9196  9201  9206  9212 
9243  9248  9253  9258  9263 

9058  9063  9069  9074  9079 
9112  9117  9122  9128  9133 
9165  9170  9175  9180  9186 
9217  9222  9227  9232  9238 
9269  9274  9279  9284  9289 

5 
5 
5 
5 
5 

85 

86 
87 
88 
89' 

9294  9299  9304  9309  9315 
9345  9350  9355  9360  9365 
9395  9400  9405  9410  9415 
9445  9450  9455  9460  9465 
9494  9499  9504  9509  9513 

9320  9325  9330  9335  9340 
9370  9375  9380  9385  9390 
9420  9425  9430  9435  9440 
9469  9474  9479  9484  9489 
9518  9523  9528  9533  9538 

5 
5 
5 
'5 
5 

9O 

91 
92 
93 
94 

9542  9547  9552  9557  9562 
9590  9595  9600  9605  9609 
9638  9643  9647  9652  9657 
9685  9689  9694  9699  9703 
9731  9736  9741  9745  9750 

9566  9571  9576  9581  9586 
9614  9619  9624  9628  9633 
9661  9666  9671  9675  9680 
9708  9713  9717  9722  9727 
9754  9759  9763  9768  9773 

5 
5 
5 
5 
5 

95 

96 
97 
98 
99 

9777  9782  9786  9791  9795 
9823  9827  9832  9836  9841 
9868  9872  9877  9881  9886 
9912  9917  9921  9926  9930 
9956  9961  9965  9969  9974 

9800  9805  9809  9814  9818 
9845  9850  9854  9859  9863 
9890  9894  9899  9903  9908 
9934  9939  9943  9948  9952 
9978  9983  9987  9991  9996 

5 

4 
4 
4 
4 

N.  B.  —  Naperian  log  =  Briggs'  log  x  2.302. 
Base  of  Naperian  System  =  e  —  2.71828. 

Trigonometric  Ratios  (Natural);  including  "arc,"  by  which  is  meant 
"radians/'  or  "7r-measure, "  or  "circular  measure ;"  e.g.,  arc  100°  =  1.7453293, 

=         Of  7T. 


de- 

arc 

gree 

sin     cosec 

tan    cotan 

sec     cos 

• 

0 

0       infin. 

0       infin. 

1.0000  1.0000 

90 

1.5703 

0.0175 

1 

0.0175  57.299 

0.0175  57.290 

L0001  0.9998 

89 

1.5533 

.0349 

2 

.0349  28.654 

.0349  28.636 

1.0006   .9994 

88 

1.5359 

.0524 

3 

.0523  19.107 

.0524  19.081 

1.0014   .9986 

87 

1.5184 

.0698 

4 

.0698  14.336 

.0690  14.301 

1.0024   .9976 

86 

1.5010 

.0873 

5 

.0872  11.474 

.0875  11.430 

1.0038   .9962 

85 

1.4835 

0.1047 

6 

0.1045   9.5668 

0.1051   9.5144 

1.0055  0.9945 

84 

1.4661 

.1222 

7 

.1219   8.2055 

.1228   8.1443 

1.0075   .9925 

83 

1.4486 

.1396 

8 

.1392   7.1853 

.1405   7.1154 

1.0098   .9903 

82 

1.4312 

.1571 

9 

.1564   6.C925 

.1584   6.3138 

1.0125   .9877 

81 

1.4137 

.1745 

10 

.1736   5.7588 

.1763   5.6713 

1.0154   .9848 

80 

1.3963 

©.1920 

11 

0.1908   5.2408 

0.1944   5.1446 

1.0187  0.9816 

79 

1.3788 

.2094 

12 

.2079   4.8097 

.2126   4.7046 

1.0223   .9781 

78 

1.3614 

.2269 

13 

.2250   4.4454 

.2309   4.3315 

1.0263   .9744 

77 

1.3439 

.2443 

14 

.2419   4.1336 

.2493   4.0108 

1.0306   .9703 

76 

1.3264 

.2618 

15 

.2588   3.8637 

.2679   3.7321 

1.0353   .9659 

75 

1.3090 

0.2793 

16 

0.2756   3.6280 

0.2867   3.4874 

1.0403  0.9613 

74 

1.2915 

.2967 

17 

.2924   3.4203 

.3057   3.2709 

1.0457   .9563 

73 

1.2741 

.3142 

18 

.3090   3.2361 

.3249   3.0777 

1.0515   .9511 

72 

1.2566 

.3316 

19 

.3256   3.0716 

.3443   2.9042 

1.0576   .9455 

71 

1.2392 

.3491 

20 

.3420   2.9238 

.3640   2.7475 

1.0642   .9397 

70 

1.2217 

0.3665 

21 

0.3584   2.7904 

0.3839   2.6051 

1.0712  0.9336 

69 

1.2043 

.3840 

22 

.3746   2.6695 

.4040   2.4751 

1.0785   .9272 

68 

1.1868 

.4014 

23 

.3907   2.5593 

.4245   2.3559 

1.0864   .9205 

67 

1.1694 

.4189 

24 

.4067   2.4586 

.4452   2.2460 

1.0946   .9135 

66 

1.1519 

.4363 

25 

.4226   2.3662 

.4663   2.1445 

1.1034   .9063 

65 

1.1345 

0.4538 

26 

0.4384   2.2812 

0.4877   2.0503 

1.1126  0.8988 

64 

1.1170 

.4712 

27 

.4540   2.2027 

.5095   1.9626 

1.1223   .8910 

63 

1.0996 

.4887 

28 

.4695   2.1301 

.5317   1.8807 

1.1326   .8829 

62 

1.0821 

.5061 

29 

.4848   2.0627 

.5543   1.8040 

1.1434   .8746 

61 

1.0646 

.5236 

30 

.5000   2.0000 

.5774   1.7321 

1.1547   .8660 

60 

1.0472 

0.5411 

31 

0.5150   1.9416 

0.6009   1.6643 

1.1666  0.8572 

59 

1.0297 

.5585 

32 

.5299   1.8871 

.6249   1.6003 

1.1792   .8480 

58 

1.0123 

.5760 

33 

.5446   1.8361 

.6494   1.5399 

1.1924   .8387 

57 

0.9948 

.5934 

34 

.5592   1.7883 

.6745   1.4826 

1.2062   .8290 

56 

0.9774 

.6109 

35 

.5736   1.7435 

.7002   1.4281 

1.2208   .8192 

55 

0.9599 

0.6283 

36 

0.5878   1.7013 

0.7265   1.3764 

1.2361  0.8090 

54 

0.9425 

.6458 

37 

.6018   1.6616 

.7536   1.3270 

1.2521   .7986 

53 

0.9250 

.6632 

38 

.6157   1.6243 

.7813   1.2799 

1.2690   .7880 

52 

0.907'j 

.6807 

39 

.6293   1.5890 

.8098   1.2349 

1.2868   .7771 

51 

0.8901 

.6981 

40 

.6428   1.5557 

.8391   1.1918 

1.3054   .7660 

50 

0.8727 

0.7156 

41 

0.6561   1.5243 

0.8693   1.1504 

1.3250  0.7547 

49 

0.8552 

7330 

42 

.6691   1.4945 

.9004   1.1106 

1.3456   .7431 

48 

0.8378 

7505 

43 

.6820   1.4663 

.9325   1.0724 

1.3673   .7314 

47 

0.8203 

.7679 

44 

.6947   1.4396 

.9657   1.0355 

1.3902   .7193 

46 

0.8028 

.7854 

45 

.7071   1.4142 

1.0000   1.0000 

1.4142   .707.1 

45 

0.7854 

cos      sec 

cotan     tan 

cosec     sin 

de- 

arc 

gree 

INDEX. 


Absolute  path  of  water  44,  57,  75,  90 

Absolute  velocity 56,  57,  58,  64 

Accumulator,  differential.  .  .  .  • '    254 

Accumulator,  hydraulic 252 

Air-compression,  hydraulic .   264,  268 

American  impulse  wheels 70 

American  turbines 130, 134 

Angular  momentum 42 

Axial-flow  turbines 113 

Back-pitch  wheels • 32 

Backwater 219, 228 

Backwater  curve, 232,  234 

Banks,  slope  of 217 

Barker's  mill 83 

Bazin's  formula  for  weirs 221 

Bell-mouthed  profiles 79 

Bernoulli's  theorem  for  a  rota- 
ting casing 56,  59,  61,  76,  98 

Boyden's  test  of  turbine 134 

Brake,  friction 149 

Bramah  press 253 

Branching  pipe 197, 199 

Breast  wheels 30 

Brotherhood  engine 249 

Bucket-engine 3 

Bucket  in  circular  path 5 

Bulk-modulus  for  water 204 

Calculations  for  pipes 188-201 

Carpenter's  experiments 260 

Cascade  impulse  wheel 70 

Centrifugal  pumps: 168-187 

best  speed 178 

diffusion-guides 179 

efficiency 179 

numerical  example 179 

practical  points 181 

Rockford 182 

starting 181 

suction-pipe 181 

theory. 174, 176 

to  maintain  fire-steam 181 

turbine  pumps 184 


Classification  of  turbines 113 

Compressed  air 264,  268 

Compressibility  of  water 204 

Conversion  scales 190 

Current-wheels 36 

Diagrams,  various,  see  Appendix. 

Differential  accumulator 254 

Diffuser 126 

Doble  impulse  wheel 70 

Doble  needle  regulating-nozzle.     72 

Draft-tube 116, 124, 127 

Efficiency : 

definition 2 

of  breast-wheels •    34 

of  Fourneyron  turbine 102 

of    Fourneyron    turbines    at 

Niagara  Falls 110 

hydraulic 181 

of  impulse  wheel 67 

of  overshots 30 

of  undershots 35 

Elasticity,  modulus  of,  for  water  204 

Elevator,  hydraulic 253 

Emerson  friction-brake 154 

Erosion;  limiting  velocities 217 

Fall  River  turbine 112 

Flat   plates   for   impulse   wheel     69 

Fly-wheel 80, 109, 167 

Foster's  hydraulic  ram 262 

Fourneyron  turbine,  theory.  .   96,97 

Fourneyron  turbines 91-112 

Fourneyron  turbines  at  Niagara 

Falls 107 

Francis  formula  for  weirs.  .    157, 158 

Francis  tests 155 

Francis  turbines 115-120 

Friction  brake 149, 154, 157 

"  Full  gate  " 95 

Gearing  of  overshots,  etc 38 

Girard  impulse  wheels 72-81 

Gravity  motor 2, 3 

Governors,  hydraulic 166 

vii 


vin 


INDEX. 


Governors,  mechanical 163 

Hazen-Williams      formula      for 

pipes 189 

Hazen-Williams  hydraulic  slide- 
rule 189 

Holyoke  testing-flume 134, 151 

Hook  gauge 150 

Hydraulic  air-compression.    264,  268 

Hydraulic  dredge. 187 

Hydraulic  grade-line .  .    191, 195,  201 
Hydraulic  motors : 

definition 1 

general  theorem  for  power  ...     13 

types  of 2 

Hydraulic  ram 257,  264 

experiments 259, 260 

Foster 262 

Mead 262 

Pearsall 262 

Phillips 263 

Rife 261 

Impulse-wheels 62,  65,  69,  70 

Inertia  motor. 2,  9 

Jack,  hydraulic 254 

Jet,  pressure  on  solid 62.  64 

Jonval  turbines 113, 121 

Joukovsky's  experiments 208 

Jump,  hydraulic 238 

Kinetic  motor 2, 9 

King  governor 163,  164 

Kutter's  formula 215 

Laxey,  overshot  wheel  at 29 

Leather  packing 255 

Leffel  turbine 135 

Lift,  hydraulic 253 

Lombard  governor 166 

Loss  of  head  in  supply -pipe .    193,  200 

Mead's  hydraulic  rani 262 

Mixed  types  of  motors 11 

Mixed-flow  turbines 113 

Modulus  of  elasticity  for  water  204 

Momentum,  angular 42 

Multistage  turbine  pumps 184 

Niagara  Falls,  turbines  at.  .   107,  116 

Nozzles  for  impulse  wheels 71 

Nozzle,  Doble  needle  regulating.     72 

Nozzle  on  pipe 193 

Open  channels,  flow  in 214,  237 

Overshot  water-wheel 22 

power  of 27 

at  Laxey 29 

Packing  for  rams,  etc 253,  255 

"  Paddle-wheel  "  as  motor 36 

Parallel-flow  turbine 121 

"  Part  gate  " 95 

Partitions    in    Fourneyron    tur- 
bines. .  95 


Pearsall's  hydraulic  ram 262 

Pelton  impulse  wheels 70,  71 

Phillips'  hydraulic  ram 263 

Pipes,  friction-head  in 188 

Pipes;  main  pipe  and  branches.     197 

Pipes,  old 189 

Pipes,  tuberculated 189 

Plates,  flat;  as  buckets 69 

Poiree's  formula L34 

Poncelet  undershot  wheels 36 

Power  lost  in  supply-pipe 202 

Power,  general  theorem  for  hy- 
draulic motor .  .  .      13 

Pressure  at  entrance  of  turbine .    145 

Pressure  of  jet  on  solid 62,  64 

"  Pressure-energy  " 8, 18 

Pressure-engines 6,  240 

Pressure-engine     with     variable 

stroke 250 

Pressure-motor 2 

Prony  friction-brake 149,     154 

Pump,     general     theorem     for 

power 19 

Pump  test 20 

Pump,  piston 242 

Pump,  water-motor 247 

Radial-flow  turbines 113 

Rafter's  experiments  on  weirs.   222 

Ram,  hydraulic 257-264 

Rankine's  formula  for  efficiency 

of  hydraulic  ram 259 

Reaction  turbine 83 

Regulating-gate  for  turbine.  ...  1 10 
Regulation  of  impulse  wheels.  .  .  71 
Regulation  by  diversion  of  jet .  .  71 
"  Relay  motor,"  for  governor. .  .  166 

Relief-valves. 211 

Rife  hydraulic  ram 261 

Rigg  engine 1  . .  .   251 

Sagebien  wheels 34 

St.  Guilhem's  formula 235 

Samson  turbine 133 

Schmidt  engine 250 

Shock,  see  Water-hammer. 

Snifting-valve 258 

Snow  governor 164 

Supply-pipe  for  turbine 200 

Swain  turbine 135 

"  Tangential  "  wheels 62 

Terni,  wheels  at 79 

Test   of  Tremont  turbine..  155, 158, 

160 

Theorems,  fundamental,  for  tur- 
bines    39,  54 

Thomson  vortex  wheel 120 

"  Throttling,"  as  means  of  regu- 
lation. .  71 


INDEX. 


IX 


Tremont  turbine 155 

Turbine  as  centrifugal  pump..    168 
Turbines : 

American 130 

axial-flow 113, 121 

Cadiat 91 

classification 113 

Combe 91 

computations  for 144 

development 91 

empirical  relations 142 

entrance  pressure 145 

Fall  River 112 

Fourneyron 91-112,  134 

general  theorem  for  power .   42,  54 

general  theory 137 

governors  for.  . 163, 166 

guides  and  vanes 141 

Hercule-Prcgres -136 

Hercules 133, 136 

Jonval 113, 121 

Leffel 133, 135 

mixed-flow 113, 130 

New  American 133, 136 

numerical  examples 103, 146 

power  of,  with  friction 55 

reaction 83 

regulation  of 162 

Risdon-Alcott 130, 132 

Samson 133 

Scotch 91 

special  feature  of 83 


Turbines: 

Swain 135 

testing 149 

Tremont,  test. 155 

Victor 132 

Whitelaw 91 

Turbine-pump 48,  60 

U  leathers 255 

Undershot  wheels 35 

Velocities,  limiting;  erosion.  ...   217 

Velocity,  absolute  and  relative .  56,  57 , 

58,  64,  75 

Velocity  of  whirl 48 

Vernayaz,  wheels  at 80 

Vortex  wheel  (Thomson) 120 

Waste  of  power  in  supply -pipe .   202 

Water,  compressibility 204 

Water,  elasticity 204 

Water-hammer 80, 203 

Water-hammer  experiments  .  .  .    208 

Water-hammer  prevention 211 

Water-motor  pump 247 

Water-power 1 

Waves,  standing 238 

Weirs 220-228 

experiments  on 222 

submerged 225 

Whirl,  velocity  of 48 

Wood,  R.  D.,  and  Co.'s  turbines  134 
Work  and  energy  theorem.  .  .    14,  52 

Working  forces.  . 1 

Worthington  water-motor-pump  247 


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